A Gaye

M Bourcier
email: bourcier@lms.polytechnique.fr

E Héripré
email: heripre@lms.polytechnique.fr

M Bornert
email: michel.bornert@enpc.fr

A Dimanov
email: dimanov@lms.polytechnique.fr

J Raphanel
email: raphanel@lms.polytechnique.fr

K Sab

QUANTITATIVE INVESTIGATION OF GRAIN BOUNDARY SLIDING IN HALITE ROCK BY SEM IN SITU TESTING AND FULL FIELD STRAIN MEASUREMENTS

An experimental investigation of the deformation mechanisms acting in halite at the scale of its polycrystalline microstructure is presented. Uniaxial compression tests on centimetric samples are performed inside the chamber of a scanning electron microscope and the obtained high definition images are processed by digital image correlation (DIC) routines. A special marking procedure provides an appropriate contrast at microscale. Standard DIC routines qualitatively demonstrate the coexistence of two deformation mechanisms: crystal slip plasticity and grain boundary sliding. A specific post-processing of the DIC-evaluated displacement fields provides a first quantification of the relative contribution of these mechanisms to overall strain and a dependence with grain size can be demonstrated. A modified DIC formulation able to separate intra-grain deformation from interfacial gliding is also proposed for a better quantification and preliminary results are presented. Such data will be of primary interest for the construction of physically-based multiscale models of the mechanical properties of halite, which might be used for security assessment of deep caverns in salt-rock.

1.

INTRODUCTION

Halite is a rock forming mineral with industrial and geotechnical applications, such as liquid or gaseous hydrocarbons storage, or more recently, energy storage in the form of compressed air. A large proportion of strategic hydrocarbon reserves are for instance stored in deep salt caverns. Experimental data on the short and long term mechanical behaviour are required for safety assessments of such underground structures. Mechanical constitutive relations currently in use are essentially based on macroscopic laboratory or field investigations, covering a necessarily limited range of time scales and loading conditions [START_REF] Bérest | Very slow creep tests on rock samples[END_REF]. In order to provide some physical foundation to the extrapolation of such data to other conditions, multiscale modelling approaches, taking into account microstructure and actual physical deformation mechanisms are of potential interest. The present work aims at providing qualititave but also quantitative information on the deformation mechanisms active in halite rock at the scale of its polycrystalline microstructure.

To do so, compression tests have been performed on synthetic halite samples in the chamber of a scanning electron microscope (SEM) at room temperature as well as at 350 • and the recorded high definition images have been processed by digital image correlation (DIC) routines. Thanks to their high spatial resolution (local strain gauge length below 2 µm for the highest used image magnification) the obtained strain maps clearly demonstrate the existence of two distinct deformation mechanisms: crystal slip plasticity (CSP), with well separated slip bands within the grains, and grain boundary sliding (GBS) at the interfaces between grains. Qualitative observations on samples with various microstructures suggest that CSP is the dominant mechanism (see also [START_REF] Carter | Creep of rocksalt[END_REF] on these slip systems), but that GBS is activated to overcome strain incompatibilities between neighbouring grains. Moreover, GBS seems to be more activ in samples with small grains than in samples with larger one, and less active when temperature increases.

The present paper aims at providing a more quantitative evaluation of the relative contribution of these two mechanisms to the overall deformation of the sample. Two approaches are proposed. A simpler one consists in a post-processing of the DIC-evaluated displacement fields with standard routines taking into account the polycrystalline structure of the material within the region of interest (ROI). A more sophisticated one makes use of a modified DIC formulation able to separate intra-grain deformation from interfacial gliding. The material and the overall experimental procedure are briefly described in next section. More information can be found in references [START_REF] Bourcier | Grain boundary sliding and crystal plasticity in halite[END_REF][START_REF] Bourcier | Étude multi-échelle des mécanismes de déformation ductiles de polycristaux de chlorure de sodium[END_REF]. The general concepts underlying the quantitative partitionning of strain into "intragrain" and "interfacial" contribution is outlined in section 3. This naturally lead to the first simpler procedure. Finally the more elaborated approach is described in the last section and preliminary results are given.

SEM IN SITU COMPRESSION TESTS ON SYNTHETIC HALITE

The material under investigation is a synthetic halite obtained by hot-pressing high purity commercial sodium chloride powder. An optional subsequent annealing allows to modify grain size. More details on these elaboration routes can be found in [START_REF] Bourcier | Grain boundary sliding and crystal plasticity in halite[END_REF][START_REF] Bourcier | Étude multi-échelle des mécanismes de déformation ductiles de polycristaux de chlorure de sodium[END_REF]. Three types of microstructures have been investigated: a coarse-grained one, with grains ranging from 200 to 500µm in size, a fine-grained one (30 to 80µm) and a bimodal one, in which small grains (about 80 µm) can be found in betwen larger ones (up to 400 µm). Samples investigated in the SEM are parallelepipedic with a typical length of 12 mm and a cross-section of about 6 × 6 mm. The lateral observation surface is dry polished. The application of DIC principles requires an appropriate local image contrast at the scale of the investiation. As polished halite samples exhibit a rather uniform grey level, an original and simple procedure has been developped to enhance this contrast without modifying the materials properties. It consists in depositing a very thin gold layer (thickness of about 30 nm) by means of standard evaporation routes for surface metallization. The metallized samples are then annealed at 450 • C under atmospheric pressure for 24 hours. During annealing, the thin gold film experiences dewetting and micron-sized gold droplets progressively form at the surface of the sample. These marker provide an efficient contrast for DIC investigation at microscale. A high magnification SEM image of these markers is provided in Fig. 1 together with a zoomed detail of a typical SEM image of a set of grains used for the investigation. Compression tests are performed with a specific test rigg that fits onto the stage of the Field Emission Gun SEM FEI Quanta 600 used in this study. Loading rate is about 8.10 -5 s -1 . The observation surface is metallized with a thin gold film in order to operate in high vacuum mode with secondary electron imaging. High definition images (4096 × 3775 pixels) with a sufficiently low noise level are recorded with a dwell time of about 20 s, leading to a total recording time for one image of about 350 s, during which the loading is stopped. A specifically designed oven allows us in addition to heat the sample. Although images are recorded at various magnifications in order to get a multiscale view of the strain field in such materials, the following presentation will be focused on the intermediate magnification with a field of view of about 1.8 × 1.5mm. An example is shown in Fig. 1c. Note that at such a magnification the marker are slightly too small (about two pixels in the diameter of a microdroplet). About 10 loading stages are considered all along the compression test. Standard DIC routines [START_REF] Bornert | Corrélation d'images[END_REF] can be used to process the obtained image sequences. In practice, the in-house code CMV has been used, with typical correlation window sizes of about 10 pixels, a grid step of similar size, an affine local transformation and a bilinear grey level interpolation (no significant difference is observed with biquintic interpolation). As relatively high strain levels are considered in this study (up to 10% macroscopic strain), DIC accuracy is not really an issue, as long as local strain associated with sufficienly large strain gauge length are considered and discontinuities in the displacement field are not considered, and will not be discussed further here. Local and averaged strains are computed according to procedures described in [START_REF] Allais | Experimental characterization of the local strain field in a heterogeneous elastoplastic material[END_REF].

A typical DIC result in the form of a strain map relative to a local gage length of about 10µm is given in Fig. 2. The overall compression strain is in this case 2% and the 2D von Mises equivalent strain (defined as 2/3(ε 2ε 1 ) where ε 2 > ε 1 are the two in-plane principal strains) is plotted; direction of compression is horizontal. A strongly heterogeneous local strain field is observed, with two types of localized deformation bands. The first ones are rectilinear and roughly parallel lines within a single grain and correspond to cristallographic slip planes, through which the displacement is discontinuous. It has been observed that the number of these lines increases with strain. Some grains undergo plasticity at rather low stress level, while other enter into plastic deformation later. The second type of deformation bands are more pronounced but less frequent: they appear at interfaces betwen grains and might not be perfectly rectilinear. They can also be observed at early stages of deformation and new bands are observed when strain increases. The comparison of strain maps observed on several samples tested under various conditions suggests that both mechanisms are required for the overall deformation of halite polycrystals. However CSP seems to be predominant, even if GBS is always present. It also appears that GBS is more generalized (i.e. most interfaces undergo GBS) in fine-grained sample, and around small grains in bimodal materials. In addition, it seem to be less present at higher temperature, where strain seem to be less heterogeneous (see [START_REF] Bourcier | Étude multi-échelle des mécanismes de déformation ductiles de polycristaux de chlorure de sodium[END_REF] for details). The following derivations aim at providing tools to quantitatively evaluate the contribution of both mechanisms to the overall strain.

STRAIN PARTITIONNING

Let Ω be the domain occupied by the whole ROI, and Ω g the disjoint surfaces occupied by the G grains g within Ω. In the absence of porosity, one has Ω = ∪ G g=1 Ω g . Let ω g = Ω g / Ω be the volume fraction of grain g within Ω. Let ξ be the displacement field over Ω.

By definition, the overall deformation gradient over Ω is F Ω = 1 + 1 |Ω| ∂Ω ξ ⊗ n dl, where 1 is the unit second order tensor and ∂Ω the boundary of Ω with outward normal n. In case ξ would only be know at discrete positions, as in the case of standard DIC routines, an approximate value of F Ω is obtained by a discretisation of the relation over the available data, and some interpolation assumption. See [START_REF] Allais | Experimental characterization of the local strain field in a heterogeneous elastoplastic material[END_REF] for the description of the obtained relations in case of a linear interpolation between neighbor measurement positions; these relations are used in CMV. In the case of a regular continuous displacement field, this contour integral can also be transformed into a surface integral involving the local displacement gradient: |Ω|(F Ω -1) = Ω ∇ξ ds.

Similar relations can be used to link the overall strain of a grain g to the displacement field at its boundary and the displacement gradient inside it: |Ω g |(F Ω g -1) = Ω g ∇ξ ds = ∂Ω p ξ ⊗ n dl. A direct consequence of the overall continuity of ξ is the following classical relation between overall strain and average strains in the grains:

F Ω = ∑ G g=1 ω g F Ω g . Note that
this relation still holds when the displacement field is no longer continuous indide the grains, as in the case of localised crystal plasticity on slip planes, but still continuous at the grain boundaries. Things are however different when a discontinuity is also observed at the grain interfaces, in which case the displacement to be used to define F Ω g is the displacement at the boundary on the grain side, say ξ g . One has thus: |Ω g |(F Ω g -1) = ∂Ω g ξ g ⊗ n g dl. Summing up all these relations for the G grains inside the ROI Ω, one gets:

∑ G g=1 |Ω g |(F Ω g -1) = |Ω|(F Ω -1) + ∑ g,g ∂Ω g ∩Ω g (ξ g ⊗ n g + ξ g ⊗ n g )ds.
The last term in this expression spans over all interfaces between grains strictly inside Ω. It does no longer vanish since ξ g might be different from ξ g even though n g = -n g . This relation can be reformulated into :

F Ω = G ∑ g=1 ω g F Ω g + 1 |Ω| W (ξ + -ξ -) ⊗ n)ds = G ∑ g=1 ω g F Ω g + 1 2 G ∑ g=1 1 |Ω| W ∩∂Ω g (ξ + g -ξ g ) ⊗ n g )ds (1)
where W is the union of all internal interfaces between grains, n its normal with arbitrarily chosen direction, ξ +ξ -is the displacement jump accross this interface along this direction, and ξ + g the displacement at the boundary of the grain g on the side opposite to the grain. This relation provides a way to separate the contributions to the overall deformation gradient of intracrystalline phenomena, such as crystal plasticity, and those due to interfacial phenomena such as GBS. The former are provided by the first sum and the second by the last one. Note that the last decomposition in Eq. 1 provides a way to define this strain partitioning at the scale of a single grain. More precisely, the intracrystalline F Intra g and the interfacial F Inter g contributions to strain of a grain g will be respectively given by

F Intra g = F Ω g = ∂Ω g ξ g ⊗ n g dl F Inter g = 1 2 ∂Ω g ξ + g ⊗ n g dl - ∂Ω g ξ g ⊗ n g dl (2)
The question is now to evaluate these quantities from experimental data. The term F Intra g can be given a first evaluation based on a discretisation of the grains interfaces on the regular grid used for the DIC analysis. In practice, its consists in attributing a "phase index" to each measurement point used in the DIC analysis (either manually, or making use of EBSD data recorded on the same sample an appropriate combination of data). Grain contours "without interface", as defined in [START_REF] Allais | Experimental characterization of the local strain field in a heterogeneous elastoplastic material[END_REF] and sketched in Fig. 3a, can be constructed and the contour integrals as defined above, provide an evaluation of F Ω g . The weighted average of these quantities quantifies the contribution of CSP to overall deformation, while the difference between this result and the average F Ω gives acces to the contribution of GBS. This procedure is a generalisation to multiphase materials of a similar one used by Rupin [START_REF] Rupin | Déformation à chaud de matériaux biphasés : modélisations théoriques et confrontations expérimentales[END_REF] in the case of two-phase steels. It has been outlined and applied in [START_REF] Bourcier | Grain boundary sliding and crystal plasticity in halite[END_REF][START_REF] Bourcier | Étude multi-échelle des mécanismes de déformation ductiles de polycristaux de chlorure de sodium[END_REF] to various halite samples, and the results confirm the above qualitative observations.

MODIFIED DIC PROCEDURE AND POST-PROCESSING

This simple procedure suffers however from two main limitations, clearly illustrated on Fig. 3a . First, used grain contours do not exactly coincide with grain boundaries and a small but non null surface of the ROI is not associated with any grain. The CSP occuring in this "interphase" will then be interpreted as GBS. Second, correlation windows relative to measurement points in these contours might overlap neighbour grains, so that the displacement evaluated at these positions might not be a good evaluation of ξ g . To overcome these difficulties, a new formulation is proposed. It makes use of similar contour integrals but based on points exactly situated on the grain interfaces. To generate such points, a regular DIC grid is first processed such that points close to an interface are translated exactly onto it (with an accuracy depending on the image resolution and the interface detection, but no longer on the DIC grid pitch). These points are represented in blue in Fig. 3c. In addition, these points will be attributed two displacements, which are evaluations of ξ g and ξ + g . To do so, DIC algorithms are modified to take into account masks of the reference image, which limit the extension of the correlation windows. These mask are constructed by means of the same labelled image (see Fig. 3b) of the microstructure as the one used to translate points near the interface. When an external mask is used (as in Fig. 4a), image contrast outside grain g is not used and the evaluated displacement will not be perturbated by GBS. On the other hand, an internal mask (Fig. 4c) avoids using information inside grains for the evaluation of the displacement of interfacial points, so that the DIC results is a good estimate of ξ + g . The corresponding strain maps (Fig. 4b andd) over the considered grain confirm that grain boundary discontinuities have not been taken into account in the first case, while they generate large strain values at some boundaries of the grain in the second one. Work is under progress to apply this procedure to the experimental data available for various microstructures.

AKNOWLEDGMENTS

The present work is part of the cooperative project "MicroNaSel" funded by the Agence Nationale de la Recherche (ANR). Support has also been provided by the chair "Énergies durables" of Ecole Polytechnique, sponsored by EDF and CEA. The FEG-SEM used in this study has been acquired with the combined financial support of Region Ile de France ("SESAME 2004" program), École polytechnique and CNRS.

6.

Figure 1 -

 1 Figure 1 -Gold droplets used as markers on halite sample: observation at high magnification (a), enlarged detail with grain interfaces (b) of a typical SEM image (c) used for DIC analysis, with ∼20 grains (coarse-grained material)

Figure 2 -

 2 Figure 2 -Typical strain map on a coarse-grained material. Equivalent strain. Two deformation mechanisms are observed. Compression axis is horizontal and average compressive strain is 2%. Local gauge length is 10µm.

Figure 3 -

 3 Figure 3 -Contours used to compute the contribution of intracristalline mechanisms to overall strain, simplified procedure (a). Labelled image (b) used to construct points on the interface (c)

Figure 4 -

 4 Figure 4 -Modified image correlation routine: external mask with DIC grid (a) and obtained strain map (b); internal mask (c) and obtained strain map (d) (arbitrary colormap)


Jonathan Chappelon

Jorge Luis

Ram Írez

Alfons Ín

Jean-Christophe Yoccoz

THE SQUARE FROBENIUS NUMBER

Keywords: . 2010 Mathematics Subject Classification. Primary 11D07 Numerical semigroups, Frobenius number, Perfect square integer

Let S = s 1 , . . . , s n be a numerical semigroup generated by the relatively prime positive integers s 1 , . . . , s n . Let k 2 be an integer. In this paper, we consider the following k-power variant of the Frobenius number of S defined as k r(S) := the largest k-power integer not belonging to S.

In this paper, we investigate the case k = 2. We give an upper bound for 2 r(S A ) for an infinite family of semigroups S A generated by arithmetic progressions. The latter turns out to be the exact value of 2 r( s 1 , s 2 ) under certain conditions. We present an exact formula for 2 r( s 1 , s 1 + d ) when d = 3, 4 and 5, study 2 r( s 1 , s 1 + 1 ) and 2 r( s 1 , s 1 + 2 ) and put forward two relevant conjectures. We finally discuss some related questions.

Introduction

Let s 1 , . . . , s n be relatively prime positive integers. Let S = s 1 , . . . , s n = n i=1 x i s i x i integer, x i 0 be the numerical semigroup generated by s 1 , . . . , s n . The largest integer which is not an element of S, denoted by g(S) or g( s 1 , . . . , s n ), is called the Frobenius number of S. It is well known that g( s 1 , s 2 ) = s 1 s 2s 1s 2 . However, calculating g(S) is a difficult problem in general. In [START_REF] Ramírez Alfonsín | Complexity of the Frobenius problem[END_REF] was shown that computing g(S) is NP-hard. We refer the reader to [START_REF] Ramírez Alfonsín | The Diophantine Frobenius Problem[END_REF] for an extensive literature on the Frobenius number.

Throughout this paper, the set of non-negative integers is denoted by N. The nonnegative integers not in S are called the gaps of S. The number of gaps of S, denoted by N(S) (that is, N(S) = #(N \ S)) is called the genus of S. We recall that the multiplicity of S is the smallest positive element belonging to S.

Given a particular (arithmetical, number theoretical, etc.) Property P , one might consider the following two P -type functions of a semigroup S: P r(S):= the largest integer having property P not belonging to S and P r(S) := the smallest integer having property P belonging to S. Notice that the multiplicity and the Frobenius number are P -type functions where P is the property of being a positive integer 1 .

In this spirit, we consider the property of being perfect k-power integer (that is, integers of the form m k for some integers m, k > 1). Let k 2 be an integer, we define k-power r(S) := the largest perfect k-power integer not belonging to S. This k-power variant of g(S) is called the k-power Frobenius number of S, we may write k r(S) for short.

In this paper we investigate the 2-power Frobenius number, we call it the square Frobenius number.

In Section 2, we study the square Frobenius number of semigroups S A generated by arithmetic progressions. We give an upper bound for 2 r(S A ) for an infinite family (Theorem 2.7) which turns out to be the exact value when the arithmetic progression consists of two generators (Corollary 2.9).

In Section 3, we present exact formulas for 2 r( a, a + 3 ) where a 3 is an integer not divisible by 3 (Theorem 3.1), for 2 r( a, a + 4 ) where a 3 is an odd integer (Theorem 3.2) and for 2 r( a, a + 5 ) where a 2 is an integer not divisible by 5 (Theorem 3.3).

In Sections 4 and 5, we turn our attention to the cases a, a + 1 where a 2 and a, a + 2 where a 3 is an odd integer. We present formulas for the corresponding square Frobenius number in the case when neither of the generators are square integers (Propositions 4.1 and 5.1). We also put forward two conjectures on the values of 2 r( a, a + 1 ) and 2 r( a, a + 2 ) in the case when one of the generators is a square integer (Conjectures 4.2 and 5.2). The conjectured values have an unexpected close connection with a known recursive sequence (Equation ( 14)) and in which √ 2 and √ 3 (strangely) appear. A number of computer experiments support our conjectures.

Finally, Section 6 contains some concluding remarks.

Arithmetic progression

Let a, d and k be positive integers such that a and d are relatively prime. Throughout this section, we denote by S A the semigroup generated by the arithmetic progression whose first element is a, with common difference d and of length k + 1, that is, S A = a, a + d, a + 2d, . . . , a + kd .

Note that the integers a, a + d, . . . , a + kd are relatively prime if and only if gcd(a, d) = 1.

We shall start by giving a necessary and sufficient condition for a square to belong to S A .

For any integer x coprime to d, a multiplicative inverse modulo d of x is an integer y such that xy ≡ 1 mod d. Proposition 2.1. Let i be an integer and let λ i be the unique integer in {0, 1, . . . , d -1} such that λ i a + i 2 ≡ 0 mod d. In other words, the integer λ i is the remainder in the Euclidean division of -a -1 i 2 by d, where a -1 is a multiplicative inverse of a modulo d. Then,

(a -i) 2 ∈ S A if and only if (i + kd) 2 i 2 + λ i a ad + k d -λ i (a + kd).
A key step for the proof of this result is the following lemma, which can be thought as a variant of a result given in [START_REF] Roberts | Note on linear forms[END_REF], see [4, Lemma 1] for a short proof. The arguments for the proof of this variant are similar to those used in the latter. Lemma 2.2. Let M be a non-negative integer and let x and y be the unique integers such that M = ax + dy, with 0 y a -1. Then, M ∈ S A if and only if y kx (with x 0).

Proof. First, suppose that M ∈ S A and let x 0 , x 1 , . . . , x k be non-negative integers such that M = k i=0 x i (a + id). Then, we have that

M = k i=0 x i a + k i=0 ix i d = x ′ a + y ′ d, with x ′ = k i=0 x i ∈ N and y ′ = k i=0 ix i ∈ N. It follows that y ′ = k i=0 ix i k k i=0 x i = kx ′ .
Moreover, since M = xa + yd with y ∈ {0, 1, . . . , a -1}, we obtain that there exists a non-negative integer λ such that

y ′ = y + λa and x ′ = x -λa.
This leads to the inequality

y = y ′ -λa kx ′ -λa = kx -λ(k + 1)a kx.
Conversely, suppose now that y kx. Obviously, since y 0, we know that x 0. Let y = qk + r be the Euclidean division of y by k, with q ∈ N and r ∈ {0, 1, . . . , k -1}. If r = 0, then we have that 0 q x since y = qk kx. It follows that

M = xa + qkd = (x -q)a + q(a + kd) ∈ S A .
Finally, if r > 0, then we have that 0 q x -1 since y = qk + r kx. It follows that

M = xa + (qk + r)d = (x -q -1)a + q(a + kd) + (a + rd) ∈ S A .
This completes the proof.

We may now prove Proposition 2.1.

Proof of Proposition 2.1. Let i be an integer and let λ i ∈ {0, 1, . . . , d -1} such that λ i a + i 2 ≡ 0 mod d. We have that

(a -i) 2 = (a -2i)a + i 2 = (a -2i -λ i )a + i 2 + λ i a d d = a -2i -λ i + i 2 + λ i a ad d a + i 2 + λ i a d - i 2 + λ i a ad a d.
We thus have, by Lemma 2.2, that the square (ai) 2 is in S A if and only if

i 2 + λ i a d - i 2 + λ i a ad a k a -2i -λ i + i 2 + λ i a ad d ⇐⇒ i 2 + λ i a d k (a -2i -λ i ) + i 2 + λ i a ad (a + kd) ⇐⇒ i 2 + λ i a kd (a -2i -λ i ) + i 2 + λ i a ad d(a + kd) ⇐⇒ i 2 + 2ikd kda -λ i (a + kd) + i 2 + λ i a ad d(a + kd) ⇐⇒ i 2 + 2ikd + k 2 d 2 kd(a + kd) -λ i (a + kd) + i 2 + λ i a ad d(a + kd) ⇐⇒ (i + kd) 2 i 2 + λ i a ad + k d -λ i (a + kd).
This completes the proof.

Remark 1. We have that λ 0 = 0 and λ i > 0 for all integers i such that gcd(i, d) = 1 with d 2. Moreover,

λ i = λ d-i for all i ∈ {1, 2, . . . , d -1}.
The above characterization permits us to obtain an upper-bound of 2 r(S A ) when a is larger enough compared to d 3.

Definition 2.3. Let λ * be the integer defined by

λ * = max 0 i d-1 λ i ∈ {0, 1, . . . , d -1} | λ i a + i 2 ≡ 0 mod d .
Let {α 1 < . . . < α n } ⊆ {0, 1, . . . , d -1} such that λ α j = λ * and take α n+1 = d + α 1 . Let j ∈ {1, . . . , n} be the index such that

(1) (µd + α j ) 2 (kd -λ * )(a + kd) < (µd + α j+1 ) 2 ,
for some integer µ 0.

Remark 2.

(a) The above index j exists and it is unique. Indeed, we clearly have that there is an integer µ such that

µd (kd -λ * )(a + kd) < (µ + 1)d. Since 0 α 1 < • • • < α n d -1, then the interval [µd, (µ + 1)d[ can be refined into intervals of the form [µd + α i , µd + α i+1 [ for each i = 1, . . . , n -1.
Therefore, there is a unique index j verifying equation ( 1). (b) We have that µd + α n+1 = (µ + 1)d + α 1 .

The following two propositions give us useful information on the sequence of indices α 1 , . . . , α n . Proposition 2.4. We have that α i + α n+1-i = d, for all i ∈ {1, . . . , n}.

Proof. Since {i ∈ {1, . . . , n} | λ i = λ * } = {α 1 , . . . , α n }, with α 1 < α 2 < • • • < α n , and since λ d-i = λ i , for all i ∈ {1, . . . , d -1}, by Remark 1. Proposition 2.5. If d 3 then n 2 and 1 α 1 < d 2 < α n d -1.
Proof. Suppose that n = 1 and hence

α n = α 1 . Since d = α 1 + α n = 2α 1 by Proposi- tion 2.4, it follows that d is even and α 1 = d 2 . If d is divisible by 4 then d 2 2 = d 4 • d ≡ 0 (mod d).
Therefore,

λ * = λ α 1 = λ d 2 = 0. Moreover, since gcd(1, d) = 1, we know that λ 1 > 0. It follows that λ 1 > λ * , in contradiction with the maximality of λ * .
If d is even, not divisible by 4, then d 2 is odd and

d 2 2 = d 2 • d 2 = d 2 -1 2 d + d 2 ≡ d 2 (mod d).
Since a is coprime to d, we know that there exists a multiplicative inverse a -1 modulo d such that aa -1 ≡ 1 mod d. Since d is even, it follows that a -1 is odd and we obtain that

λ d 2 ≡ -a -1 d 2 2 ≡ -a -1 d 2 ≡ d 2 (mod d).
Therefore,

λ d 2 = d 2 . Moreover, for any i ∈ 0, . . . , d 2 -1 , since i + d 2 2 = i 2 + id + d 2 2 ≡ i 2 + d 2 (mod d)
and since a -1 is odd, it follows that

λ i+ d 2 ≡ -a -1 i + d 2 2 ≡ -a -1 i 2 -a -1 d 2 ≡ λ i + d 2 (mod d),
for all i ∈ 0, . . . , d 2 -1 . Since d 3, we have that 1

< d 2 < 1 + d 2 < d. Finally, since λ 1 > 0, we deduce that max λ 1 , λ 1+ d 2 > d 2 = λ d 2 ,
in contradiction with the maximality of λ d

2

. We thus have that if d 3 then n 2 and α 1 < α n . Since α 1 + α n = d, by Proposition 2.4, we deduced that α 1 < d 2 < α n . This completes the proof. Definition 2.6. Let us now consider the integer function h(a, d, k) defined as

h(a, d, k) := (a -((µ -k)d + α j+1 )) 2 .
Remark 3. We notice that the function h(a, d, k) can always be computed for any relatively prime integers a and d and any positive integer k. It is enough to calculate λ i for each i = 0, . . . , d -1, from which λ * and the set of α i 's can be obtained and thus the desired µ and α j+1 can be computed. We need the following lemma before proving Theorem 2.7.

Lemma 2.8. If d 3 then α i+1 -α i d -1 and α i + α i+1 2d
for all i ∈ {1, . . . , n}.

Proof. First, let i ∈ {1, . . . , n -1}. Since 1 α j d -1, for all j ∈ {1, . . . , n}, from Remark 1, it follows that

α i+1 -α i < α i+1 d -1 and α i + α i+1 < 2d.
Finally, for i = n, since n 2 and α n > α 1 by Proposition 2.5, it follows that

α n+1 -α n = d + α 1 -α n < d.
Moreover, since α n = dα 1 by Proposition 2.4, we obtain that

α n + α n+1 = (d -α 1 ) + (d + α 1 ) = 2d.
This completes the proof.

We now have all the ingredients to prove Theorem 2.7.

Proof of Theorem 2.7. It is known [START_REF] Roberts | Note on linear forms[END_REF] that

g(S A ) = a -2 k + 1 a + (d -1)(a -1) -1. Since a 2 > a-2 k + 1 a, 2akd > (d -1)(a -1) and (kd) 2 > 0 then g(S A ) < a 2 + 2kda + (kd) 2 = (a -(-kd)) 2 .
Therefore, it is enough to show that (ai) 2 ∈ S for all -kd i < (µk)d + α j+1 .

We have two cases.

Case 1. -kd i (µ -k)d + α j . We have that (i + kd) 2 (µd + α j ) 2 (since i (µ -k)d + α j ) (kd -λ * )(a + kd) (by definition) i 2 +λ i a ad + k d -λ i (a + kd) (since λ * λ i and i 2 +λ i a ad 0)
Therefore, by Proposition 2.1, we obtain that (ai) 2 ∈ S A .

Case 2. (µ -k)d + α j < i < (µ -k)d + α j+1 .
In this case we have that α j < i mod d < α j+1 implying that λ i λ * -1 and thus

(2) (kd -λ i )(a + kd) (kd -λ * )(a + kd) + (a + kd).
Moreover,

(i + kd) 2 < (µd + α j+1 ) 2 (3) = ((µd + α j ) + (α j+1 -α j )) 2 = (µd + α j ) 2 + (α j+1 -α j ) (2 (µd + α j ) + (α j+1 -α j )) = (µd + α j ) 2 + (α j+1 -α j ) (2µd + α j + α j+1 ) . Now, from Lemma 2.8, we have that (4) α ℓ+1 -α ℓ < d and α ℓ + α ℓ+1 2d
for all ℓ ∈ {1, . . . , n}. Therefore, combining (3) and (4), we obtain

(5) (i + kd) 2 < (µd + α j ) 2 + d (2µd + 2d) = (µd + α j ) 2 + 2d 2 (µ + 1)
for a j ∈ {1, . . . , n}. Since

(µd + α j ) 2 (by def inition) (kd -λ * ) (a + kd) (2)
(kd

-λ i ) (a + kd) -(a + kd) then (6) (i + kd) 2 < (kd -λ i ) (a + kd) + 2d 2 (µ + 1) -(a + kd).
We claim that (7) 2d 2 (µ + 1) a + kd.

We have two subcases Subcase i) For j ∈ {1, . . . , n -1}. Since (µd + α j ) 2 (kdλ * )(a + kd) < (µd + α j+1 ) 2 , α j 1 and α j+1 α n < d then

µ = (kd -λ * )(a + kd) d .
Moreover, since a

+ kd 4kd 3 > 4(kd -λ * )d 2 , it follows that (8) µ 2(kd -λ * ) with λ * > 0.
If µ = 2(kdλ * ), then we have

2d 2 (µ + 1) = 4kd 3 + 2(1 -2λ * )d 2 4kd 3 a + kd, as announced. Otherwise, if µ > 2(kd -λ * ), it follows that (kd -λ * )(a + kd) (µd + α j ) 2 (αj 1) > µ 2 d 2 > µ 2 -1 d 2 = (µ -1) (µ + 1) d 2 2(kd -λ * )(µ + 1)d 2 ,
obtaining the claimed inequality (7) for j ∈ {1, 2, . . . , n -1}.

Subcase ii) For j = n. Since (µd

+ α n ) 2 (kd -λ * )(a + kd) < (µd + α n+1 ) 2 , where α n = d -α 1 and α n+1 = d + α 1 , we obtain ((µ + 1)d -α 1 ) 2 (kd -λ * )(a + kd) < ((µ + 1)d + α 1 ) 2 . Since α 1 < d, we have (kd -λ * )(a + kd) d ∈ {µ, µ + 1} .
Moreover, since a + kd 4kd 3 > 4(kdλ * )d 2 , it follows that (9) µ + 1 2(kdλ * ).

If µ + 1 = 2(kdλ * ), then we have

2d 2 (µ + 1) = 4(kd -λ * )d 2 < 4kd 3 a + kd, obtaining the claimed inequality (7). Otherwise, if µ + 1 > 2(kd -λ * ), since α 1 < d 2 from Proposition 2.5, we obtain (kd -λ * )(a + kd) ((µ + 1)d -α 1 ) 2 > (µ + 1)d - d 2 2 = µ 2 + µ + 1 4 d 2 > µ (µ + 1) d 2 µ 2(kd-λ * ) 2(kd -λ * )(µ + 1)d 2 ,
obtaining the claimed inequality (7) when j = n. Finally, since inequality (7) is true for any j ∈ {1, . . . , n} then, from equation (6) we have

(i + kd) 2 < (kd -λ i ) (a + kd) + 2d 2 (µ + 1) -(a + kd) (kd -λ i ) (a + kd).
We deduce, by Proposition 2.1, that (ai) 2 ∈ S A . This completes the proof.

Remark 4. The above proof can be adapted if we consider the weaker condition a + kd > 4(kdλ * )d 2 + d 2 instead of a + kd 4kd 3 .

We believe that the upper bound h(a, d, k) of 2 r(S A ) given in Theorem 2.7 is actually an equality. We are able to establish the latter in the case when k = 1 for any d 3. Let i = (µ -1)d + α j+1 . We have

i 2 = ((µ -1)d + α j+1 ) 2 = ((µd + α j ) -(d + α j -α j+1 )) 2 = (µd + α j ) 2 -(d + α j -α j+1 ) (2 (µd + α j ) -(d + α j -α j+1 )) = (µd + α j ) 2 -(d + α j -α j+1 ) ((2µ -1)d + α j + α j+1 ) .
Since d + α jα j+1 1, by Lemma 2.8, and (2µ -1)d + α j + α j+1 > (2µ -1)d, it follows that (10)

i 2 < (µd + α j ) 2 -(2µ -1)d (d -λ * )(a + d) -(2µ -1)d.
Since a + d 4d 3 , we already know that µ + 1 2(dλ * ) (see equations ( 8) and ( 9) with k = 1). It follows that µ 2(dλ * ) -1 1 and then

(11) d -λ * µ + 1 2 2µ -1.
By combining equations ( 10) and (11) we obtain

i 2 < (d -λ * )(a + d) -(d -λ * )d = (d -λ * )a and i 2 + λ i a ad = i 2 + λ * a ad < (d -λ * )a + λ * a ad = 1.
We may thus deduce that i 2 + λ i a ad = 0.

Finally, since

(i + d) 2 = (µd + α j+1 ) 2 > (d -λ * )(a + d) = i 2 + λ i a ad + 1 d -λ i (a + d),
we deduce, from Proposition 2.1, that (ai) 2 ∈ a, a + d , as desired.

Unfortunately, the value of 2 r( a, a + d ) given in the above corollary does not hold in general (if the condition a + d 4d 3 is not satisfied). However, as we will see below, the number of values of a not holding the equality 2 r( a, a + d ) = h(a, d, 1) is finite for each fixed d.

Formulas for a, a + d with small d 3

In this section, we investigate the value of 2 r( a, a + d ) when d is small. For any positive integer d 3, we may define the set E(d) to be the set of integers a coprime to d not holding the equality of Corollary 2.9, that is, E(d) := a ∈ N \ {0, 1} gcd(a, d) = 1 and 2 r( a, a + d ) = h(a, d, 1) .

Since λ *

d -1 then, from Corollary 2.9, we obtain that E(d) ⊂ [2, 4d 3 -1] ∩ N. We completely determine the set E(d) for a few values of d 3 by computer calculations, see Table 1. The exact values of 2 r( a, a + d ) when a ∈ E(d), for d ∈ {3, . . . , 12}, are given in Appendix A.

d |E(d)| E(d) 3 0 ∅ 4 0 ∅ 5 5 {2,
For each value d ∈ {3, . . . , 12}, an explicit formula for 2 r( a, a + d ) can be presented excluding the values given in Table 1. The latter can be done by using (essentially) the same arguments as those applied in the proofs of Theorem 2.7 and Corollary 2.9. We present the proof for the case d = 3. Theorem 3.1. Let a 2 be an integer not divisible by 3 and let S = a, a + 3 . Then, 2 and a ≡ 1 mod 3 or (3b + 1) 2 2(a + 3) < (3b + 2) 2 and a ≡ 2 mod 3, (a -(3b + 1)) 2 if either (3b + 2) 2 a + 3 < (3b + 4) 2 and a ≡ 1 mod 3 or (3b + 2) 2 2(a + 3) < (3b + 4) 2 and a ≡ 2 mod 3.

2 r(S) =            (a -(3b -1)) 2 if either (3b + 1) 2 a + 3 < (3b + 2)
Proof. Since g(S) = (a -1)(a + 2) -1 = a 2 + a -3 < (a + 1) 2 then 2 r(S) (a -1) 2 .
By Proposition 2.1, we know that

(12) (a -i) 2 ∈ S ⇐⇒ (i + 3) 2 3 i 2 + λ i a 3a + 3 -λ i (a + 3),
where λ i ∈ {0, 1, 2} such that λ i a + i 2 ≡ 0 mod 3, that is,

λ i =  
 0 if i ≡ 0 mod 3 and a ≡ 1, 2 mod 3, 1 if i ≡ 1, 2 mod 3 and a ≡ 2 mod 3, 2 if i ≡ 1, 2 mod 3 and a ≡ 1 mod 3.

We have four cases.

Case 1. Suppose that a ≡ 1 mod 3 with (3b + 1)

2 a + 3 < (3b + 2) 2 . Note that b 1 since a + 3 19. If i 3b -2 then (i + 3) 2 (3b + 1) 2 a + 3 3 i 2 + λ i a 3a + 3 -λ i (a + 3),
obtaining, by equation ( 12), that (ai) 2 ∈ S.

If i = 3b -1 then i 2 = (3b -1) 2 = 9b 2 -6b + 1 b 1 < 9b 2 + 6b -2 = (3b + 1) 2 -3 a, obtaining that 0 i 2 + λ i a 3a = i 2 + 2a 3a < 1 (since 3b -1 ≡ 2 mod 3)
and thus

i 2 + λ i a 3a = 0.
Moreover, since

3 i 2 + λ i a 3a + 3 -λ i (a + 3) = a + 3 < (3b + 2) 2 = (i + 3) 2 ,
by equation ( 12), we have that (ai) 2 / ∈ S.

Case 2. Suppose that a ≡ 1 mod 3 with (3b + 2)

2 a + 3 < (3b + 4) 2 . If b = 0, we have (a -1) / ∈ S since 3 1 + λ i a 3a + 3 -λ i (a + 3) = a + 3 < 4 2 .
Therefore 2 r(S) = (a -1) 2 in this case. Suppose now that b 1. If i 3b -1, then

(i + 3) 2 (3b + 2) 2 a + 3 3 i 2 + λ i a 3a + 3 -λ i (a + 3),
obtaining, by equation ( 12), that (ai) 2 ∈ S.

If i = 3b then, using that λ i = 0,

(i + 3) 2 = (3b + 3) 2 3(3b + 2) 2 3(a + 3) 3 i 2 + λ i a 3a + 3 -λ i (a + 3),
obtaining, by equation ( 12), that (ai) 2 ∈ S.

If i = 3b + 1 then i 2 = (3b + 1) 2 = 9b 2 + 6b + 1 b 1 < 9b 2 + 12b + 1 = (3b + 2) 2 -3 a, obtaining that 0 i 2 + λ i a 3a = i 2 + 2a 3a < 1 (since 3b + 1 ≡ 1 mod 3)
and thus i 2 + λ i a 3a = 0.

Moreover, since

3 i 2 + λ i a 3a + 3 -λ i (a + 3) = a + 3 < (3b + 4) 2 = (i + 3) 2 ,
therefore, by equation ( 12), we have that (ai) 2 / ∈ S.

Case 3. Suppose that a ≡ 2 mod 3 with (3b + 1) 2 2(a + 3) < (3b + 2) 2 . Note that b 1 since 2(a + 3) 16. If i 3b -2 then (i + 3) 2 (3b + 1) 2 2(a + 3)

λ i 1 3 i 2 + λ i a 3a + 3 -λ i (a + 3),
obtaining, by equation ( 12), that (ai) 2 ∈ S.

If i = 3b -1 then i 2 = (3b -1) 2 = 9b 2 -6b + 1 b 1 < 9b 2 + 6b -5 = (3b + 1) 2 -6 2a, obtaining that 0 i 2 + λ i a 3a = i 2 + a 3a < 1 (since 3b -1 ≡ 2 mod 3)
and thus

i 2 + λ i a 3a = 0.
Moreover, since

3 i 2 + λ i a 3a + 3 -λ i (a + 3) = 2(a + 3) < (3b + 2) 2 = (i + 3) 2 ,
therefore, by equation ( 12), we have that (ai) 2 / ∈ S.

Case 4. Suppose that a ≡ 2 mod 3 with (3b + 2) 2 2(a + 3) < (3b + 4) 2 . If b = 0, we have (a -1) / ∈ S since

3 1 + λ i a 3a + 3 -λ i (a + 3) = 2(a + 3) < 4 2 .
Therefore 2 r(S) = (a -1) 2 in this case. Suppose now that b 1. If i 3b -1 then

(i + 3) 2 (3b + 2) 2 2(a + 3) 3 i 2 + λ i a 3a + 3 -λ i (a + 3),
obtaining, by equation ( 12), that (ai) 2 ∈ S.

If i = 3b then, using that λ i = 0,

(i + 3) 2 = (3b + 3) 2 b 1 < 3 2 (3b + 2) 2 3(a + 3) 3 i 2 + λ i a 3a + 3 -λ i (a + 3).
Therefore, by equation ( 12), we have (ai) 2 ∈ S.

If i = 3b + 1 then i 2 = (3b + 1) 2 = 9b 2 + 6b + 1 b 1 < 9b 2 + 12b -2 = (3b + 2) 2 -6 2a,
when b 1 and clearly i 2 = 1 < 2a when b = 0, obtaining that

0 i 2 + λ i a 3a = i 2 + a 3a < 1 and i 2 + λ i a 3a = 0.
Moreover, since

3 i 2 + λ i a 3a + 3 -λ i (a + 3) = 2(a + 3) < (3b + 4) 2 = (i + 3) 2 ,
therefore, by equation( 12), we have that (ai) 2 / ∈ S.

The proofs of the following two theorems are completely analogous to that of Theorem 3.1 with a larger number of cases to be analyzed (in each case, the appropriate inequality is obtained in order to apply Proposition 2.1). Theorem 3.2. Let a 3 be an odd integer and let S = a, a + 4 . Then,

2 r(S) =              (a -(4b -1)) 2 if either (4b + 1) 2 a + 4 < (4b + 3) 2 and a ≡ 1 mod 4
or (4b + 1) 2 3(a + 4) < (4b + 3) 2 and a ≡ 3 mod 4, (a -(4b + 1)) 2 if either (4b + 3) 2 a + 4 < (4b + 5) 2 and a ≡ 1 mod 4 or (4b + 3) 2 3(a + 4) < (4b + 5) 2 and a ≡ 3 mod 4.

Theorem 3.3. Let a 2 be an integer not divisible by 5 and let S = a, a + 5 . Then,

2 r(S) =                                                1 if a = 2 or 4, 10 2 if a = 13, (a -6) 2 if a = 27 or 32, (a -(5b -2)) 2 if either (5b + 2) 2 a + 5 < (5b + 3) 2
and a ≡ 4 mod 5 or (5b + 2) 2 2(a + 5) < (5b + 3) 2 and a ≡ 2 mod 5, (a -(5b -1)) 2 if either (5b + 1) 2 a + 5 < (5b + 4) 2 and a ≡ 1 mod 5 or (5b + 1) 2 2(a + 5) < (5b + 4) 2 and a ≡ 3 mod 5, a = 13, (a -(5b + 1)) 2 if either (5b + 4) 2 a + 5 < (5b + 6) 2 and a ≡ 1 mod 5 or (5b + 4) 2 2(a + 5) < (5b + 6) 2 and a ≡ 3 mod 5, (a -(5b + 2)) 2 if either (5b + 3) 2 a + 5 < (5b + 7) 2 and a ≡ 4 mod 5, a = 4 or (5b + 3) 2 2(a + 5) < (5b + 7) 2 and a ≡ 2 mod 5, a = 2, 27, 32.

Study of a, a + 1

We investigate the square Frobenius number of a, a + 1 with a 2. We first study the case when neither a nor a + 1 is a square integer. Proof. Since g( a, a + 1 ) = a 2a -1 then (a -1) 2 g( a, a + 1 ) < a 2 .

We thus have that 2 r( a, a + 1 ) < a 2 . We shall show that (ai) 2 ∈ a, a + 1 for i ∈ {1, 2, . . . , b -1}.

We first observe that

(13) (a -i) 2 = a 2 -2ai + i 2 = (a -2i)a + i 2 = (a -2i -i 2 )a + i 2 (a + 1),
for any integer i.

Since for any i ∈ {1, 2, . . . , b -1} we have

a -2i -i 2 = a -i(i + 2) a -(b -1)(b + 1) = a -b 2 + 1 > 0
and i 2 > 0 then, by (13), we deduce that (ai) 2 ∈ a, a + 1 for any i ∈ {1, 2, . . . , b -1}.

Finally, since a + 1 < (b + 1) 2 (implying that a -2bb 2 < 0) and 0 < b 2 < a then we may deduce, from (13), that (ab) 2 ∈ a, a + 1 .

Let (u n ) n 1 be the recursive sequence defined by ( 14) . . . , see [START_REF]The On-line Encyclopedia of Integers Sequences[END_REF]. We pose the following conjecture in the case when either a or a + 1 is a square integer.

u 1 = 1, u 2 = 2, u 3 = 3, u 2n = u 2n-1 + u 2n-
r( a, a + 1 ) =            a -b √ 2 2 if b ∈ n 0 {u 4n+1 , u 4n+2 } , a -b √ 3 2 if b ∈ n 0 {u 4n+1 , u 4n+2 } . If a + 1 = b 2 for some integer b 1 then 2 r( a, a + 1 ) =                    a -b √ 2 2 if b ∈ n 1 {u 4n-1 , u 4n } , a -b √ 3 2 if b ∈ n 1 {u 4n , u 4n+3 } , 2 2 if b = u 3 = 3.
The formulas of Conjecture 4.2 have been verified by computer for all integers a 2 up to 10 6 .

Study of a, a + 2

We investigate the square Frobenius number of a, a + 2 with a 3 odd. We first study the case when neither a nor a + 2 is a square integer. Proposition 5.1. Let a 3 be an odd integer such that (2b + 1) 2 < a < a + 2 < (2b + 3) 2 for some integer b 1. Then, Proof. Since g( a, a + 2 ) = (a -1)(a + 1) -1 = a 2 -2 then (a -1) 2 < g( a, a + 2 ) < a 2 .

We thus have that 2 r( a, a + 2 ) < a 2 . We shall show that (ai) 2 ∈ a, a + 2 for i ∈ {1, 2, . . . , 2b}.

We first observe that for any integer i, we have

(15) (a -2i) 2 = a 2 -4ai + 4i 2 = (a -4i)a + 4i 2 = (a -4i -2i 2 )a + 2i 2 (a + 2).
Since for any i ∈ {1, 2, . . . , b} we have a -4i -2i 2 = a -2i(i + 2) a -2i(2i + 1) > a -(2i + 1) 2 a -(2b + 1) 2 > 0 and 2i 2 > 0 then, by (15), it follows that (a -2i) 2 ∈ a, a + 2 for any i ∈ {1, 2, . . . , b}.

Moreover, for any integer i, we have

(16) (a -(2i + 1)) 2 = a 2 -2a(2i + 1) + (2i + 1) 2 = (a -2(2i + 1))a + (2i + 1) 2 = (a -4i -3)a + (2i + 1) 2 + a = a -4i -3 -(2i+1) 2 +a 2 a + (2i+1) 2 +a 2 (a + 2) = a-4i 2 -12i-7 2 a + (2i+1) 2 +a 2 (a + 2) = a+2-(2i+3) 2 2 a + (2i+1) 2 +a 2 (a + 2).
Note that a + 2 -(2i + 3) 2 and (2i + 1) 2 + a are even because a is odd.

Since, for any i ∈ {0, 1, . . We pose the following conjecture in the case when either a or a + 2 is a square integer. The formulas of Conjecture 5.2 have been verified by computer for all odd integers a 3 up to 10 6 .

Concluding remarks

In the process of investigating square Frobenius numbers different problems arose. We naturally consider the P -type function k-power r(S) = k r(S) defined as, Problem 1. Let k 2 be an integer and let S be a numerical semigroup. Investigate the computational complexity to determine k r(S) and/or k r(S).

Or more ambitious, Question 1. Let k 2 be an integer. Is there a closed formula for k r(S) and/or k r(S) for any semigroup S?

Perhaps a first step on this direction might be the following.

Problem 2. Give a formula for 2 r( F i , F j ) and/or 2 r( F i , F j ) with gcd(F i , F j ) = 1 where F k denotes the k th Fibonacci number. What about 2 r( a 2 , b 2 ) where a and b are relatively prime integers ? We clearly have that 2 r( a 2 , b 2 ) = a 2 for 1 a < b.

Theorem 2 . 7 .

 27 Let d 3 and a + kd 4kd 3 . Then, 2 r(S A ) h(a, d, k) .

Corollary 2 . 9 .

 29 Let d 3 and a + d 4d 3 . Then, 2 r( a, a + d ) = h(a, d, 1) . Proof. By Theorem 2.7, we have 2 r( a, a + d ) (a -((µ -1)d + α j+1 )) 2 . It is thus enough to show that (a -((µ -1)d + α j+1 )) 2 ∈ a, a + d .

Proposition 4 . 1 .

 41 Let a be a positive integer such that b 2 < a < a + 1 < (b + 1) 2 for some integer b 1. Then, 2 r( a, a + 1 ) = (ab) 2 .

Conjecture 4 . 2 .

 42 Let (u n ) n 1 be the recursive sequence given in (14). If a = b 2 for some integer b 1 then

2

 2

2 r

 2 ( a, a + 2 ) = (a -(2b + 1)) 2 .

Conjecture 5 . 2 . 2 r 2 {u 4n+1 } , 38 2 if

 52222 Let (u n ) n 1 be the recursive sequence given in (14). If a = (2b + 1) 2 for some integer b 1 then if 2b + 1 = u 5 = 7.If a + 2 = (2b + 1) 2 for some integer b 1 then 2 r( a, a + 2 ) = (2b + 1) ∈ n 0 {u 4n+3 } , a -(2b + 1) √ 3 2 if (2b + 1) ∈ n 0 {u 4n+3 } .

  k r(S) := the smallest perfect k-power integer belonging to S. It is clear that (17) s k r(S) s k where s is the multiplicity of S. Theorem 6.1. Let S A = a, a + d, . . . , a + kd where a, d, k are positive integers with gcd(a, d) = 1. If d ak 1+2k then 2 r(S A ) (ad) 2 . Proof. We shall use the characterization given in Proposition 2.1 with i = d. In this case λ d = 0 and d ak 1+2k < a thus d 2 + 0a ad + k d -0 (a + kd) = d a + k d -0 (a + kd) = akd + (kd) 2 . Thus, (d + kd) 2 akd + (kd) 2 ⇐⇒ d 2 + 2kd 2 akd ⇐⇒ d ak 1 + 2k . Therefore, by Proposition 2.1, (ad) 2 ∈ S A .

Table 1 .

 1 E(d) for the first values of d 3.

	4, 13, 27, 32}

  2 and u 2n+1 = u 2n + u 2n-2 for all n 2. The first few values of (u n ) n 1 are 1, 2, 3, 5, 7, 12, 17, 29, 41, 70, 99, 169, 239, 408, 577, 985, . . . . . .

	This sequence appears in a number of other contexts. For instance, it corresponds to the denominators of Farey fraction approximations to √ 2, where the fractions are 1 1 , 2 1 , 3 2 , 4 3 , 7 5 , 10 7 , 17 12 , 24 17

The second author was partially supported by INSMI-CNRS.

Appendix A. Complement to formulas for a, a + d with small d 3

In Tabular 2, we compare the exact values of 2 r( a, a + d ) and the formula h(a, d, 1), when a ∈ E(d) for d ∈ {3, . . . , 12}.


Michel Abgrall
email: michel.abgrall@obspm.fr

Jocelyne Guéna

Michel Lours

Giorgio Santarelli

E Tobar

Sébastien Bize

Serge Grop

Benoît Dubois

Christophe Fluhr

Vincent Giordano

G Santarelli ⇤ †

M E Tobar

High Stability Comparison of Atomic Fountains using two Different Cryogenic Oscillators

niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

I. INTRODUCTION

In almost two decades, the number and the performances of cold atom fountains [START_REF] Guéna | Progress in atomic fountains at LNE-SYRTE[END_REF], [START_REF] Wynands | Atomic fountain clocks[END_REF], [START_REF] Li | Improved accuracy of the NPL-CsF2 primary frequency standard: Evaluation of distributed cavity phase and microwave lensing frequency shifts[END_REF], [START_REF] Heavner | First accuracy evaluation of NIST-F2[END_REF] have drastically increased, allowing the improvement of international and national time scales, and of the realization of the second of the international system of units (SI). These frequency standards have also many applications such as the characterization of new time and frequency transfer techniques, absolute frequency measurements and fundamental physics tests.

At SYRTE, we operate an ensemble of three fountains, the two caesium fountains FO1 and FOM and the dual fountain FO2 operated simultaneously with caesium and rubidium atoms. These fountains regularly provide calibrations that are sent to the BIPM to participate to the steering of TAI, the international atomic time, and are also used to produce the new version of the French time scale UTC(OP) since 2012 [START_REF] Rovera | The New UTC(OP) based on LNE-SYRTE Atomic Fountains[END_REF], [START_REF] Abgrall | Performances of UTC(OP) based on LNE-SYRTE atomic fountains[END_REF], [START_REF] Guéna | Contributing to TAI with a Secondary Representation of the SI Second[END_REF]. Initially these fountains were compared using a common reference based on a BVA quartz oscillator phase locked to an hydrogen maser. The measurement stability was limited to ⇠ 10 13 , due to the phase noise of the oscillator because of the Dick effect [START_REF] Santarelli | Frequency stability degradation of an oscillator slaved to a periodically interrogated atomic resonator[END_REF]. This oscillator was replaced by an ultra stable reference based on a cryogenic sapphire oscillator (CSO) with liquid helium refill developed by the University of Western Australia (UWA) [START_REF] Luiten | Power stabilized cryogenic sapphire oscillator[END_REF], [START_REF] Mann | Cryogenic sapphire oscillator with exceptionally high frequency stability[END_REF]. This allows the fountains to operate at the fundamental quantum projection noise limit [START_REF] Santarelli | Quantum Projection Noise in an Atomic Fountain: A High Stability Cesium Frequency Standard[END_REF], thus reducing the averaging time needed for improving the accuracy evaluation. Our CSO is in continuous operation at SYRTE since about ten years. It presents a short term stability of 2 3 ⇥ 10 15 at a 1 s integration time. Using this CSO and an ensemble of low noise synthesizers [START_REF] Chambon | Design and realization of a flywheel oscillator for advanced time and frequency metrology[END_REF], [START_REF] Chambon | Design and metrological features of microwave synthesizers for atomic fountain frequency standard[END_REF] as a reference for FO2-Cs, we have demonstrated the best stability ever obtained with a microwave frequency standard at the level of 1.6 ⇥ 10 14 ⌧ 1/2 , where ⌧ is the averaging period.

Nowadays, new developments provide alternatives to refilled liquid helium CSO for producing ultra stable microwave references. The principle of microwave generation from the optical domain based on a frequency comb laser stabilised on an ultra stable laser has been demonstrated in [START_REF] Millo | Ultralow noise microwave generation with fiber-based optical frequency comb and application to atomic fountain clock[END_REF], [START_REF] Weyers | Reaching the quantum limit in a fountain clock using a microwave oscillator phase locked to an ultrastable laser[END_REF]. In the experiment we performed at SYRTE, we demontrated a stability of 2 ⇥ 10 15 when comparing the output signal of our CSO to the signal produced by the stabilized optical frequency comb [START_REF] Millo | Ultralow noise microwave generation with fiber-based optical frequency comb and application to atomic fountain clock[END_REF]. On the other hand, new generations of CSOs using pulse-tube cryocoolers are under development in different laboratories [START_REF] Hartnett | Ultra-low vibration pulse-tube cryocooler stabilized cryogenic sapphire oscillator with 10 16 fractional frequency stability[END_REF], [START_REF] Hartnett | Cryogenic sapphire oscillator using a low-vibration design pulse-tube cryocooler: first results[END_REF], [START_REF] Grop | ELISA: An ultrastable oscillator for ESA deep space antennas[END_REF], [START_REF] Grop | ELISA: A cryocooled 10 GHz oscillator with 10 15 frequency stability[END_REF], [START_REF] Nand | Resonator power to frequency conversion in a cryogenic sapphire oscillator[END_REF], demonstrating the improvement of atomic fountain stability [START_REF] Takamizawa | Atomic Fountain Clock With Very High Frequency Stability Employing a Pulse-Tube-Cryocooled Sapphire Oscillator[END_REF]. This paper presents the results of an experiment performed in collaboration with FEMTO-ST institute that develops stateof-the-art cryocooled CSO systems. For a few months, one of their CSO, named Uliss, was operated at SYRTE. At FEMTO-ST institute, this CSO has demonstrated a short term stability of 6 7 ⇥ 10 16 at 1 s integration time remaining below 1 ⇥ 10 15 between 1 and 10 4 s when compared to a similar device [START_REF] Grop | Latest Improvements in the Performances of a Cryogenic Sapphire Oscillator[END_REF]. Uliss was also used as a low noise frequency reference for the ground tests of the PHARAO cold atom space clock [START_REF] Ph | Tests on ground of the flight model of the PHARAO cold atom space clock[END_REF] (CNES, Toulouse, France), and for a deep space network station [START_REF] Solana | Design of the F&T Subsystem for ESA's Deep Space Antenna 3[END_REF] (ESA DSA 3 station, Malargüe, Argentina). The level of performances of this CSO also allowed to characherize microwave generation from the optical domain based on a solid state frequency comb [START_REF] Dolgovskiy | Ultra-stable microwave generation with a diode-pumped solid-state laser in the 1.5-m range[END_REF]. This article presents the characterisation at SYRTE of two different ultra stable references, one based on our UWA CSO (named Molly) [START_REF] Hartnett | Cryogenic sapphire oscillator with exceptionally high long-term frequency stability[END_REF], [START_REF] Tobar | Long term operation and performance of cryogenic sapphire oscillators[END_REF] and the second one based on the FEMTO-ST CSO Uliss, each of them either free running or phase locked to the same hydrogen maser (H-maser). We further demonstrate the performances of these devices in a comparison between two fountains, namely the caesium and the rubidium parts of FO2 operated simultaneously (FO2-Cs and FO2-Rb) [START_REF] Guéna | Demonstration of a Dual Alkali Rb/Cs Fountain Clock[END_REF], each one being referenced to one of the two CSOs.

II. MEASUREMENT SETUP

The measurement set-up is shown in Figure 1. The operation of the two CSOs is based on a cylindrical monocrystalline sapphire resonator oscillating in the so called whispering gallery modes. The crystals are maintained at a low temperature of about 6 K, corresponding to the turning point of the resonance frequency slightly above liquid helium temperature. The quality factor of the resonators are of about 1⇥10 9 . In both devices, the cavity is housed in a copper insert installed in a cryogenic system. For Molly, it is a 250 l dewar that has to be liquid helium refilled every 26 days, whereas for Uliss, the system is based on a pulse-tube cryocooler that can be operated without interruptions. The resonance frequency is about 11.932 GHz for Molly and 9.988 GHz for Uliss. The oscillation of both CSOs is maintained using a Pound-Galani servo loop, including phase, amplitude and power servo loops.

Both CSOs are equipped with ultra low noise synthesizers in order to produce reference signals phase locked to a Hmaser, which allows to benefit both from the short term stability of the free running CSOs and from the long term stability of the H-maser.

The synthesizers of Molly have been described in detail in [START_REF] Chambon | Design and realization of a flywheel oscillator for advanced time and frequency metrology[END_REF] and [START_REF] Chambon | Design and metrological features of microwave synthesizers for atomic fountain frequency standard[END_REF]. A first synthesizer produces coherent 11.98 GHz and 100 MHz signals that are phase locked to the H-maser with a time constant of a few 1000 s using a Direct Digital Synthesizer (DDS). This phase lock is based on a 100 MHz beatnote. A second synthesizer using these signals produces a 1 GHz reference signal.

Similarly, the Uliss synthesizer [START_REF] Grop | Frequency synthesis chain for ESA deep space network[END_REF] produces a 10 GHz signal phase locked to the H-maser using a DDS, with a time constant also of a few 1000 s. A signal at 1 GHz is also generated.

In next sections, we present the characterization of the signals generated by these two different ultra stable references in terms of phase noise and frequency stability. Furthermore, we have checked the effect of the cryocooler pulse-tube on the phase of the signal generated by Uliss that could produce a frequency shift of the atomic fountain. We also verified the spectrum of the delivered signal in order to check the level of sidebands. We finally present the direct comparison between two fountains each one referenced to one of the CSOs. During this measurement, as shown in Figure 1, FO2-Cs was operated as usual, using its 9.192 GHz synthesis based on the 11.98 GHz provided by Molly, whereas the 6.834 GHz synthesis of FO2-Rb was generated using the 1 GHz signal produced by Uliss. generating these references (for Molly, 11.932 GHz against 11.98 GHz, blue curve; for Uliss, 10 GHz against 9.988 GHz, magenta curve). We have also verified the noise of the 6.834 GHz FO2-Rb synthesizer referenced on Uliss (red curve).

III. CHARACTERIZATION RESULTS

A. Phase noise

Because there is a ⇠ 2 GHz frequency difference between the two CSOs, we had to implement a specific measurement system to compare the free running oscillators (see Figure 3). The two signals at 11.932 GHz and 9.988 GHz are mixed together and then down converted to the RF domain using two stages of mixing with a 1 GHz signal. The 1 GHz signal was provided by Molly, but no significant difference was observed in the results when we used the 1 GHz from Uliss. The resulting RF signal at ⇠56 MHz is finally mixed to the output signal of a ⇠23 MHz DDS after a frequency doubling stage. The DDS is clocked using a 100 MHz signal provided by Molly. The DDS was tuned in order to set the beatnote signal at the quadrature before being sent to a FFT analyser.

The same setup was adapted for the comparison between the two phase locked oscillators at 11.98 GHz and 10 GHz: we used the same two stages of down conversion, leading to a beatnote at 20 MHz. The beatnote was adjusted to the quadrature after mixing to the output of the DDS set to 20 MHz.

To characterize the CSOs synthesizers, we mixed the free running oscillator output to the phase locked output (11.932 GHz vs 11.98 GHz and 9.988 GHz vs 10 GHz for Molly and Uliss, respectively). The resulting RF signal was then mixed to the DDS set to the appropriate frequency: ⇠ 48 MHz for Molly and ⇠ 12 MHz for Uliss.

The phase noise of the 6.834 GHz FO2-Rb synthesizer referenced to the Uliss 1 GHz signal was measured using the beatnote obtained in the comparison against a backup synthesizer at the same frequency referenced to the Molly 1 GHz signal. All these measurements were performed using a HP 3561A FFT analysing the beatnote signals adjusted to the quadrature. We verified that we obtain similar results using a Symmetricom 5125A signal analyser.

The plot of the measurement results in Figure 2 shows a phase noise of -90 dBrad 2 /Hz at 1 Hz from the carrier for the comparison between the two free running CSOs (black curve), and better than -120 dBrad 2 /Hz for Fourier frequencies beyond a few kHz. This corresponds to an Allan deviation of about 2 3 ⇥ 10 15 at 1 s integration time. The Uliss synthesizer noise (magenta curve) presents a flicker noise of -95 f 1 dBrad 2 /Hz between 1 Hz and a few kHz. The bump at higher frequencies is due to the lock bandwidth of the 2.5 GHz DRO used by the system but remains below -120 dBrad 2 /Hz. The Molly synthesizer is slightly better for Fourier frequencies below 100 Hz (blue curve), but the phase lock loop of the 11.98 GHz DRO produces a bump of about -105 dBrad 2 /Hz at about 1 kHz. These results are similar to what was obtained in 2005, when the synthesizer was put in operation [START_REF] Chambon | Design and realization of a flywheel oscillator for advanced time and frequency metrology[END_REF], except for the DRO PLL gain that we have not tried to re-optimize, since the performances remained sufficiently good for application to the fountains.

The comparison between the 11.98 GHz and the 10 GHz signals of the two CSOs phase locked to the H-maser presents a phase noise equivalent to that of the free running CSOs for frequencies below 100 Hz. For higher frequencies, it is limited by the Molly synthesizer. The phase noise measured at 6.834 GHz is equivalent to a flicker noise of -87 f 1 dBrad 2 /Hz between 1 Hz and 1 kHz and limited by the backup synthesizer for higher frequencies. This is equivalent to a frequency noise of ⇠ 5 ⇥ 10 15 at 1 s integration time for the combination of both synthesizers. This will produce a negligible contribution to the stability of FO2-Rb, as will be seen in section IV. The red and blue curves correspond to comparisons Uliss vs H-maser and Molly vs H-maser, respectively, measured using a 100 MHz phase comparator (Timetech PCO). The stability at 1 s is about 8⇥10 14 , because in actual practice this is not the direct 100 MHz output of the H-maser that is distributed to the CSOs, but the output of a BVA/VCXO 100 MHz synthesizer phase locked to the H-maser. The bump at a few seconds corresponds to the time constant of the phase lock loop of the BVA quartz oscillator to the H-maser. The stability decreases down to below 10 16 for an averaging period of about 1 day, demonstrating the good operation of the phase lock loop of both CSOs to the H-maser. Due to this phase lock, a small bump is observed at averaging periods of about 3000 s. We note that this effect is slightly smaller for Uliss than for Molly because of a lower frequency drift when free running. The red and blue squares on the graph, obtained while analysing the Phase noise of the comparison between the two free running CSOs (black), of the CSO synthesizers (blue for Molly, pink for Uliss), of the comparison between the two phase locked CSOs (green), and of the comparison between two 6.834 GHz synthesizers referenced to 1 GHz signals provided by Molly and Uliss, respectively, to be used by FO2-Rb (red). The results are given at the measurement frequency, without rescaling. The green curves in Figures 4 and5 give the comparison between the 100 MHz outputs of the two CSOs as measured by the PCO. It is limited by the instrument noise until averaging periods of a few hundred seconds (noise floor of about 2 ⇥ 10 14 ⌧ 1 ).

B. Stability

The purple curve reproduced in Figures 4 and5 gives the Allan deviation obtained in the comparison between the 11.98 GHz and the 10 GHz signals from Molly and Uliss, respectively. This measurement was performed with the setup described in Figure 3 adapted for the stability characterization: the ⇠ 20 MHz DDS was adjusted to get a beatnote at about 65 Hz and the output of the last mixer is AC coupled using a 4 Hz bandpass filter centred at this frequency. The resulting beatnote frequency is measured using a frequency counter (Agilent model 53132A). We also performed several tests using the 5125A Symmetricom signal analyzer measuring directly the ⇠ 20 MHz beatnote and obtained the same results. The comparison of the microwave outputs of the CSOs shows a stability at 1 s as low as 2 3⇥10 15 and reaches 6⇥10 16 for averaging periods of about 100 s. For longer integration times the stability increases to about 10 15 , due to the bump of the phase lock loop, and decreases back to lower than 10 16 for averaging periods of 1 day, as for the measurements performed with the 100 MHz signals.

The magenta curve in Figure 5 shows the stability of the comparison between the two free running CSOs (11.932 GHz vs 9.988 GHz). It was measured with the previous setup except that the DDS was set to ⇠ 23 MHz and then frequency doubled. We obtain the same short term stability as for the comparison 11.98 GHz vs 10 GHz of the two CSOs phase locked to the H-maser, for averaging times up to a few tens of seconds. For longer periods the stability is degraded by the frequency drift of the CSOs, dominated by that of Molly. This is in agreement with the analysis of the frequency corrections applied to the CSOs synthesizers.

C. Phase transient

We have characterized the effect of the cryocooler pulsetube of Uliss on the output signal phase. This verification was important because there could have been a phase transient transmitted to the interrogating signal of the fountain that could produce an additional frequency shift. This characterization was performed using the nominal and backup 6.834 GHz synthesizers of FO2-Rb referenced to the 1 GHz outputs of Uliss and Molly, respectively. We used the heterodyne phase transient analyser described in [START_REF] Santarelli | Switching atomic fountain clock microwave interrogation signal and high-resolution phase measurements[END_REF] regularly operated to verify the absence of synchronous perturbations on our fountains. This device allows to average the phase of the signal synchronously to the pulse-tube of the cryocooler. For this test, the trigger was produced from one of the internal temperature sensors of Uliss, that is sensitive enough to observe thermal variations inside a cycle of the cryocooler pulse-tube: we used one of the analog output of the resonator temperature servo loop that has been amplified.

Figure 6 shows the results of two phase transient measurements obtained after averaging over about 1 day, separated by several months. The duration of the measurement window is about 1.4 s, which corresponds to two cycles of the pulse-tube. We clearly see an impact of the resonator thermal transient on the signal, which can reach several tens of µrad (see black curve in Figure 6). We observed that the shape and the amplitude of these transients varied in time, probably because of variations in the pulse-tube regime. Such phase variations could lead to frequency shifts of the fountain clock of several 10 15 if they were synchronous to the fountain cycle time. Fortunately, because FO2-Rb cycle time is about 1.6 s, the perturbation will average down to zero. This level of perturbation will also have a negligible impact on the frequency stability of the fountain, that is at the level of a few 10 14 ⌧ 1/2 .

D. Spurious characterization

We have also checked the spectral purity of the 6.834 GHz output of the FO2-Rb synthesiser referenced to Uliss using a 1 Hz resolution spectrum analyser. This has been done with the synthesizer connected to the fountain to be as representative as possible of the fountain operation. The main spurious lines are at a frequency of 100 Hz and its harmonics. As can be seen in Figure 7 these sidebands are 65 dB below the carrier and quite

IV. FOUNTAIN OPERATION

To fully demonstrate the operation of the cryocooled CSO as a fountain reference, we have performed a frequency comparison between FO2-Rb operated using its synthesizer referenced to Uliss, against FO2-Cs operated as usual, using its interrogating signal based on Molly's reference. During these measurements, both CSOs were phase locked to the same Hmaser. Figure 8 presents the overlapping Allan deviations that we have obtained after 28 consecutive days of measurement. The measurement was interrupted only a half day for a liquid helium refill of Molly. The red and blue dotted curves give the individual stability of each fountain after averaging the data over 100 s. For FO2-Rb, the stability is the same as when operated using Molly as reference. The graph shows short term stabilities of 4.6 ⇥ 10 14 ⌧ 1/2 for FO2-Rb Uliss and 4.3⇥10 14 ⌧ 1/2 for FO2-Cs Molly. These stabilities include the fact that we alternate between full and half atomic density in order to evaluate the effect of cold collisions in real time, which corresponds to fully accurate operation of the fountains. If we take full density measurements only (red and blue lines), the short term stability is improved to 3.5 ⇥ 10 14 ⌧ 1/2 and 3.4 ⇥ 10 14 ⌧ 1/2 for the two comparisons, respectively. The observed stabilities present only a small bump at about 3000 s due to the phase lock loop of the CSOs to the H-maser. The Allan deviations reach the 10 16 range after an averaging time of ⇠ 10 4 s. For averaging periods higher than a few days, the Allan deviations show the frequency drift of the H-maser, that is about 10 16 /d. This is removed if we calculate the differences of the fountain data over synchronous intervals, as can be seen on the black squares Allan deviation curve that reaches 2 ⇥ 10 16 after averaging over 5 days. The difference FO2-Cs FO2-Rb averaged over the total measurement duration is 1.6⇥10 16 . This result is fully compatible with the total systematic uncertainty of 4⇥10 16 from the fountain accuracy budgets, and agrees with measurements performed before and after, using Molly as the reference for both fountains.

V. CONCLUSION

This paper presents a full characterization of two different type ultrastable sapphire oscillator references, one based on a refilled liquid helium CSO and the second using a pulse-tube cryocooled CSO, both phase locked to the same H-maser. We have measured the phase noise and the Allan deviation of the oscillators either free running or stabilized, reaching a short term stability at the level of 2 3 ⇥ 10 15 at 1 s. We have also verified that the performances of the associated synthesizers are consistent with this result. We have checked the operation of the cryocooled CSO in terms of phase transient and spectral purity that could have produce systematics in a fountain operation. Furthermore we have demonstrated the operation of two fountains each one referenced to one of the two CSOs leading to a short term stability of ⇠ 3.5 ⇥ 10 14 ⌧ 1/2 when operated at high atomic density. At last, we have performed a 28 days quasi continuous fountain frequency comparison reaching a stability of 2⇥10 16 in fully accurate operation, including cold collisions evaluation. The result was in complete agreement with the accuracy budgets of the fountains and with earlier and later comparisons using the SYRTE CSO for both fountains.

At SYRTE, we are currently improving the long term operation of ultrastable microwave generation based on an optical frequency comb stabilized to an ultrastable laser as a backup reference oscillator to our CSO for the fountains operation. This work performed in collaboration with FEMTO-ST Institute demonstrates that a cryocooled CSO can also provide an alternative for this kind of applications.

Figure 2

 2 Figure2summarizes the different phase noise measurements that we have performed on the microwave signals generated by the two CSOs. We have characterized the noise of the two free running oscillators(11.932 GHz of Molly against 9.988 GHz of Uliss, black curve), of the two microwave references phase locked to the H-maser(11.98 GHz of Molly against 10 GHz of Uliss, green curve), and of the synthesisers

Fig. 1 .

 1 Fig. 1. Measurement setup. The two parts of FO2 using Cs and Rb cold atoms issued from 2DMOTs operate simultaneously with same cycle time.

Figures 4

 4 Figures 4 and 5 show a summary of the various stability measurements performed using Uliss and Molly. All the curves correspond to overlapping Allan deviations measured in a bandwidth of a few Hz.

Fig. 2 .

 2 Fig. 2.Phase noise of the comparison between the two free running CSOs (black), of the CSO synthesizers (blue for Molly, pink for Uliss), of the comparison between the two phase locked CSOs (green), and of the comparison between two 6.834 GHz synthesizers referenced to 1 GHz signals provided by Molly and Uliss, respectively, to be used by FO2-Rb (red). The results are given at the measurement frequency, without rescaling.

Fig. 3 .

 3 Fig. 3. Measurement system for phase noise and stability characterization of the microwave outputs of the two CSOs.

Fig. 4 .

 4 Fig. 4. Stability characterization of Uliss CSO, as measured either in the microwave domaine, or using a 100 MHz phase comparator (PCO), and as analysed from the inloop error and the DDS corrections.

Fig. 5 .

 5 Fig. 5. Stability characterization of Molly CSO, as measured either in the microwave domaine, or using a 100 MHz phase comparator (PCO), and as analysed from the inloop error and the DDS corrections.

Fig. 6 .

 6 Fig. 6. Two examples of phase transients observed on the 6.834 GHz signal referenced to Uliss averaged synchronously to the pulse-tube cycle time.

Fig. 7 .

 7 Fig. 7. Spectrum of the 6.834 GHz FO2-Rb synthesizer referenced to Uliss 1 GHz signal.

  symmetric. This corresponds to a frequency shift well below 10 16 .

Fig. 8 .

 8 Fig. 8. Stabilities of two fountains referenced to the two CSO based systems: stability of FO2-Rb referenced to Uliss and of FO2-Cs referenced to Molly operated either at full atomic density or in alternate mode (full/half atomic density) to evaluate the cold collisions; stability of the comparison between FO2-Cs and FO2-Rb.

ACKNOWLEDGMENT SYRTE is UMR CNRS 8630 between Centre National de la Recherche Scientifique (CNRS), Université Pierre et Marie Curie (UPMC) and Observatoire de Paris. LNE, Laboratoire National de Métrologie et d'Essais, is the French National Metrology Institute.


Dr Julien Roger
email: julien.roger@u-bourgogne.fr

Dr Jean-Cyrille Hierso
email: jean-cyrille.hierso@u-bourgogne.fr

Phenol Derivatives in Ruthenium-Catalyzed C-H Arylation: a General Synthetic Access to Azole-Based Congested Polyaromatics

Aryl triflates and related phenolates are suitable electrophile coupling partners for the ruthenium-catalyzed direct arylation of heteroaromatic substrates using azole N-directed sp 2 -C-H activation. We report herein convenient conditions in the efficient ortho-C-H functionalization of aryl-pyrazoles, thiazoles and pyridines where [RuCl2(p-Cym)]2 precatalyst is employed with pivalic acid (PivOH) as co-catalyst. Different phenolate derivatives were successfully coupled which tolerate a large scope of electron-rich substituents in para-, meta-and highly hindered ortho-position. Electron-withdrawing aryl triflates were found less reactive, making the general reactivity of these electrophiles complementary to those of aryl chlorides and deactivated bromides. This cost-effective ruthenium C-H activation/arylation synthesis of poly(hetero)aromatics was concurrently examined using triflates, mesylates, sulfonates and carbonates, and was also successfully extended to the use of diethyl carbonate as an eco-friendly solvent.

Introduction

Selective C-H activation/arylation reactions have resulted in the development of valuable strategies to form aromatic molecules in straightforward and atom-economic protocols. [1] Further development of general synthetic conditions which are compatible with cost-efficiency and sustainable chemistry is highly desirable. For instance, the use of more eco-friendly solvents at lower temperatures is pertinent. [2] This approach may be advantageously combined with the exploitation of coupling partners obtained from renewable resources. In this context, investigation on the use of electrophile alternatives to haloarenes, obtained for instance from alcohols is highly appropriate. Many alcohols are directly available from bio-resources, and the preparation of reactive alcohol derivatives is generally easy to achieve. Accordingly, because of their wide availability, rather low cost and practical protection abilities, phenols are frequently used in total synthesis. The phenolic group can be used to direct and introduce the desired functionality into aromatic rings, and then can achieve carbon-carbon bond formation via the corresponding sulfonate.

In this regard, palladium-catalyzed C-H functionalization has successfully exploited sulfonates which are prepared from phenolic materials, providing thus coupling reagents that are crystalline and fairly stable towards hydrolysis. [3] The use of aryl trifluoromethyl sulfonates in ruthenium-based catalysis is limited to date to very few examples. The relatively low-cost of ruthenium as transition metal is yet very attractive for cost-efficient industrial applications. [4] Phenyl triflate as electrophilic reagent was reported by Oi et al. in the presence of [RuCl2(C6H6)]2 and PPh3 in NMP and led to a mixture 61:7 of mono/diarylated phenyl pyridine. [4a] 5-Aryltetrazole was monoarylated with [RuCl2(p-Cym)]2 and MesCO2H or the amino acid N-pivaloyl-L-valine (Piv-Val-OH) as co-catalyst in toluene in the presence of aryl bromides. Three examples of aryl triflates were reported under such conditions. [4b,c] Weix and coworkers reported the coupling of 2-methylbenzoic acid with phenyl triflate using [RuCl2(p-Cym)]2 and 4,4'-di-tertbutyl-2,2'-dipyridyl (dtbbpy) as ligand in NMP with a moderate 58% yield. [4d] Similar yields for the same substrates were obtained in the presence of the cationic [Ru(tBuCN)6](BF4)2, with KOC(CF3)3, in tBuCN at 140°C. [4e] Clearly, general protocols for efficient N-ligand directed orthoarylation of aromatics using sulfonates is still lacking. Based on our recent works in using azole ligands [5] for palladium-catalyzed aromatic halogenation reactions [6] we devised conditions for the exploitation of aryl trifluoromethylsulfonates as valuable electrophile coupling partners for aromatic azoles by using ruthenium-catalyzed N-ligands directing C-H activation/arylation. The catalytic system combining [RuCl2(p-Cym]2 and pivalic acid promotes C-H functionalization of a wide range of highly functionalized phenolate derivatives using pyrazoles, benzothiazoles and pyridines as directing groups. We introduced additionally an eco-friendly protocol employing diethyl carbonate as solvent in ruthenium catalytic arylation with triflates.

Results and Discussion

Aryl triflates as coupling partners for phenyl-1H-pyrazole Poly(hetero)aromatic and biphenyl motifs are valuable synthetic scaffolds in contemporary chemistry with regards to their wide application as pharmaceuticals, agrochemicals, and in material sciences. [7] Biphenyls bind to a wide range of proteins with high levels of specificity with antihypertensive, antithrombotic antirheumatic, anti-inflammatory and analgesic properties. [8] In the convergent construction of these scaffolds the research focus has shifted to direct C-H metal-catalyzed functionalization of arenes. The development of methods using cost-effective ruthenium-catalyzed C-H arylation has provided attractive routes to classical organometallic cross-coupling approaches. [1c,d] Accordingly, our screening experiments started with the coupling of phenyl-1H-pyrazole and phenyl triflate, both commercially available (Table 1). We achieved coupling by using [RuCl(p-Cymene)]2 in toluene in the presence of KOAc. In the absence of ruthenium phenyl-1H-pyrazole was found to be unreactive in refluxing toluene (Table 1, entry 1). Conversely, in the presence of 2.5 mol% of [RuCl2(p-Cym)]2 (i.e. 5 mol% of [Ru]), a good conversion of pyrazole starting material (87%) yielded an unsatisfactory mixture of 24% of the biphenyl 1a and 63% of diarylated 1b (entry 2).

Table 1. Ruthenium catalysed coupling of phenyl-1H-pyrazole with phenyl triflate (Scheme 1). [a] entry additive (mol%) solvent T °C base (equiv.)

Conv.

(%)

1a (%) 1b (%)
1 [b] - [d] 9 [e] PivOH ( 30 PivOH: pivalic acid. [b] No ruthenium catalyst. [c] Pre-catalyst [Ru(OPiv)2(p-Cym)] (2.5 mol%). [d] Isolated yield. [e] Pre-catalyst [RuCl2(p-Cym)]2: 1.25 mol%.

Trifluoromethylbenzene, used as solvent, was found more effective with a 96% conversion of pyrazole giving however only 67% of 1b (entry 3). Using 1,4-dioxane did not improve the conversion or the selectivity (entry 4). [Ru(OPiv)2(p-Cym)] previously used for ortho-arylation of functionalized arenes with aryl chlorides [START_REF] Ackermann | For carboxylates used in ruthenium-catalyzed C-H arylations[END_REF] essentially provided a much lower selectivity with a 42:48 mixture of 1a and 1b (entry 5). Changing the base for K2CO3 did not improve this selectivity (42% of 1a, entry 6). In the presence of N-acetyl-L-valine (Ac-Val-OH) amino acid a notable improvement was achieved with a 74% yield obtained for 1b (entry 7). Finally, high conversion and selectivity in diarylated 1b was achieved by using 30 mol% of PivOH as additive (entry 8). A lower catalyst loading of 1.25 mol% of [RuCl2(p-Cym)]2 (2.5 mol% of [Ru]), or a lower temperature (80 °C) clearly limited the efficiency of the catalyst in the ortho-directed diarylation (entries 9 and 10, respectively). Interestingly, monoarylation was preferentially achieved by tuning of the heteroaryl/triflate ratio (2/1) thus yielding 70% of 1a SI).

With the optimized conditions in hands, we investigated the scope of functional aryl triflates tolerated in this coupling with phenyl-1H-pyrazole (Table 2).

Table 2. Diarylation of phenyl-1H-pyrazole from functional aryl triflates. [a] entry Major product Conv.

(%) [b] Selectivity (%) [b] Yield (%) [c] 1 100 96 89 6 100 99 [d] 95 7 100 90 [d] 83 8 100 48 [d] 38 9 100 99 [d] 85 [a] Conditions: phenyl-1H-pyrazole (1.0 equiv.), aryl triflate (3.0 equiv.), [RuCl2(p-Cym)]2 (2.5 mol%), PivOH (30 mol%), K2CO3 (4.0 equiv.), PhCF3 (0.125 M), 110 °C, under argon, 22 h. [b] Determined by 1 H NMR based on the phenyl-1H-pyrazole. [c] Isolated yield. [d] Pre-catalyst [RuCl2(p-Cym)]2 (5 mol%), PivOH additive (60 mol%).

Electron-donating alkyl substituents placed in para-position of the aryl triflate, such as methyl and isopropyl groups, gave polyaromatic 2b (89%) and 3b (72%) in very good isolated yields (Table 2, entries 1-2). 4-Acetylphenyl triflate was also easily coupled and 4b was obtained with a high isolated yield of 96% (Table 2, entry 3). Electron-poor aryl triflates were found significantly less reactive and 5 mol% of [RuCl2(p-Cym)]2 with 60 mol% of pivalic acid were necessary to isolate the monoarylated 5a from 2-pyridyl triflate in 25% yield (Table 2, entry 4). 4-In the same conditions, nitrophenyl triflate did not react with phenyl-1Hpyrazole and was recovered unchanged after 22 h. On the other hand, congested meta-substituted aryl triflates were efficiently coupled and 2-naphthyl triflate reacted with phenyl-1H-pyrazole to give 6b in 90% isolated yield (entry 5). 3,6-Dimethoxyphenyl triflate was found more demanding and by doubling the catalyst loading of [Ru]/PivOH catalyst diarylated 7b was isolated in excellent 95% yield (entry 6). Ortho-substituted aryl triflates were also used for the formation of highly congested (hetero)polyaromatic 8b-10b (Table 2, entries 7-9). By using 5 mol% of the [RuCl2(p-Cym)]2 and 60 mol% of pivalic acid, the ortho-methylated 8b was isolated in excellent 90% yield (entry 7). The formation of ortho-methoxylated 9b was more difficult (formed in 48% together with the monoarylated 9a in 52%) and isolated yields of 9b and 9a were moderate (38% and 42%, respectively, entry 8). High conversion to the pentaphenyl 1-(2,6di(naphthalene-1-yl)phenyl)-1H-pyrazole 10b was achieved with a high 85% isolated yield (entry 9).

N-directing heteroaryl derivatives as coupling partners for aryl triflates

We further applied these general coupling conditions to other heteroaryl substrates which incorporate functions at the aromatic or heteroaromatic moieties. By using the bulky 2-naphthyl triflate coupling partner we investigated the direct arylation of variously substituted arylpyrazoles, pyridines and thiazoles (Table 3). Using 2.5 mol% of [RuCl2(p-Cym)]2 with 30 mol% of PivOH, and K2CO3 in trifluoromethylbenzene at 110 °C, the 1-(4trifluoromethylphenyl)-1H-pyrazole gave diarylated 11b in 79% isolated yield with a 87% selectivity (Table 3, entry 1).

Table 3. Ruthenium-catalyzed N-directing heteroaryl derivatives coupling to 2naphtyl triflate. [a] entry substrate Major product Conv. Selectivity Yield (%) [b] (%) [b] (%) [c] 1 100 13/87 79 2 nd [d] nd 43

3 [e] 26 100/0 15 4 nd [d] nd 55 5 100 13/87 81 6 [f] 63 17/83 40 [a] Conditions: heteroaryl (1.0 equiv.), 2-naphthyl triflate (3.0 equiv.), [RuCl2(p-Cym)]2 (2.5 mol%), PivOH (30 mol%), K2CO3 (4.0 equiv.), PhCF3 (0.125 M based on phenyl-1H-pyrazole), 110 °C, under argon, 22 h. 1 H NMR yield based on the heteroaryl. [b] NMR yield. [c] Isolated yield. [d] Not determined since several side-products co-exist. [e] 2-naphthyl triflate (1.5 equiv), [RuCl2(p-Cym)]2 (5 mol%), PivOH (60 mol%), K2CO3 (2.0 equiv.). [f] [RuCl2(p-Cym)]2 (5 mol%), PivOH (60 mol%).

The coupling of 4-chlorophenyl pyrazole was found to be more challenging, giving a moderate 43% isolated yield of 12b (entry 2). The lack of selectivity is possibly due to competitive oxidative addition of chloride to ruthenium. From 3-nitrophenyl pyrazole arylation occurs selectively in the para-position from nitro substituent, giving a modest 15% yield of monoarylated 13a (entry 3, see also Table S1 in SI). Conversely, 2-(4-chloro-1H-pyrazol-1-yl)-phenyl coupled with 2-naphthyl triflate to give 14b in 55% isolated yield (entry 4). Thus, a functional group on the N-directing pyrazole unit was tolerated in ruthenium catalysis, while we have recently shown that similar palladium ortho-C-H functionalization from substituted pyrazole directing groups is a very challenging issue. [9a] These coupling were successfully extended to C-H ortho-functionalization of 2-phenylpyridine with 2-naphthyl triflate, which furnished 15b in 85% isolated yield (entry 5), and to 2phenylbenzothiazole that gave 16b in very good 83% yield (40% after workup, entry 6).

Coupling in dichloroethane (DCE) solvent of aryl triflates and heteroaryles

Ruthenium-catalyzed arylation using N-directing ligands have been mostly performed in solvents such as NMP or 1,4-dioxane, which are considered as poor eco-friendly solvents regarding waste issues (incineration, recycling, bio-treatments and VOC emissions) and toxicity (reprotoxicity, mutagenicity). [2b] Toluene and its derivatives, such as xylene and trifluoromethylbenzene are up to now considered less harmful and might be recommended as valuable media alternatives. Progress in ruthenium-catalyzed C-H arylation has been achieved by using aryl chloride electrophile coupling partners in water, [10] and by the employment of tosylates first in NMP, then in water and solventfree conditions. [11] We envisioned that sustainable conditions could be also reached by the employment of aryl triflate derivatives in diethylcarbonate (DCE) as solvent. [12] We tested our present catalytic protocol in this purpose. By using DCE further optimization appeared to be necessary, and we satisfactorily coupled aryl triflates with N-directing arylpyrazole and arylpyridine after careful conditions screening (Table S2 in SI).

Scheme 1. Ruthenium-catalyzed triflate coupling in diethyl carbonate. [a] In the presence of [RuCl2(p-Cym)]2 and KOAc in DCE at 120 °C, we obtained 1b in 60% after 48 h (Scheme 1). The coupling of aryl triflates with phenyl-1H-pyrazole was extended to yield the polyaromatic methylated 2b, acetylated 4b, naphthylated 6b and methoxylated 17b with fairly good to excellent yield (49%, 91%, 92% and 73% respectively). The catalytic system also successfully achieved 4-tolyl triflate coupling with 1-(4trifluoromethylphenyl)-1H-pyrazole to give 18b in 93% yield, and the coupling with 2-phenylpyridine gave 19b in 55% yield.

General reactivity of phenolate coupling partners

The general reactivity and comparative adequacy of various phenolates in C-C coupling is a question generally poorly addressed while it may be decisive in the efficiency of catalytic processes. Herein, we comparatively tested different leaving groups derived from 2-naphthol under our general conditions. Phenol derivatization is easy to handle and we synthesized at gram scale (up to 2 g) five sulfonate, carbonate and acetate reagents. The introduction of trifluoromethane sulfonate group was performed under anhydrous conditions using trifluoromethane sulfonic anhydride to give 20a in 91% isolated yield (Table 4, entry 1). Similarly, we achieved the synthesis of tosylate 20b, mesylate 20c, carbonate 20d and acetate 20e in 85% to 99% yield (Table 4, entries 2-5). Naphthalen-2-yl 4methylbenzenesulfonate 20b reacted with phenyl-1H-pyrazole to give diarylated 6b in high 93% yield (Scheme 2, conditions A). Interestingly our protocol in PhCF3 was found very efficient since related studies using tosylates have been limited to single arylation reactions. [11a,11b] The mesylate derivative was found slightly less reactive but achieved a very good 87% yield (Scheme 2, conditions A). A limitation of our protocol was reached with the coupling of carbonates since ethyl naphthalen-2-yl carbonate 20d achieved only a limited conversion to a mixture of 6a and 6b (5% and 10%, respectively). Finally, naphthalen-2-yl acetate 20e did not react under these conditions.

1 [a] Tf2O (1.2) -OSO2CF3 20a, 91
2 [b] ClSO2Tolyl (3.0) -OSO2Tolyl 20b, 90

3 [c] ClSO2CH3 (3.0) -OSO2CH3 20c, 85 4 [d] ClCO2C2H5 (3.0) -OSO2C2H5 20d, 91

5 [e] ClCOCH3 (3.0) -OCOCH3 20e, 99

Conditions: [a] 2-naphthol (20, 1 equiv.), NEt3 (1.5 equiv.), trifluoromethane sulfonic anhydride (1.2 equiv.) in dry CH2Cl2 at 0°C; [b] ) 2-naphthol (20, 1 equiv.), NEt3 (3.0 equiv.), p-toluenesulfonyl chloride (3.0 equiv.) in CH3CN at RT; [c] 2naphthol (20, 1 equiv.), NEt3 (3.0 equiv.), methanesulfonyl chloride (3.0 equiv.) in CH3CN at RT; [d] 2-naphthol (20, 1 equiv.), NEt3 (3.0 equiv), ethyl chloroformate (3.0 equiv.) in CH2Cl2 at RT; [e] 2-naphthol (20, 1 equiv.), NEt3 (3.0 equiv), acetyl chloride (3.0 equiv.) in CH2Cl2 at RT.

In DEC solvent slower reactions were achieved for naphthalen-2-yl sulfonate 20b and 20c (Scheme 2, conditions B),) to give after 48 h a mixture of unreacted phenyl-1H-pyrazole reagent, monoarylated 6a and diarylated 6b in 19/33/48 ratio.

Conclusions

We reported general conditions for selective ortho-diarylation of various arylheteroaryl substrates by using N-ligand directed ruthenium-catalyzed coupling of highly functionalized aryl phenolate derivatives. The complex [RuCl2(p-Cym)]2 combined with pivalic acid (PivOH), allows very efficient coupling in ortho C-H functionalization of aryl triflates, which overcome the chemoselectivity issues related to mixtures of mono and diarylation products. Our general protocol tolerated electrondonating substituents in para-, meta-and ortho-position of aryl triflates, including significantly congested substituents. Functionalized arylpyrazoles, pyridines and thiazoles were also tolerated, while this is known to be rather difficult in related palladium C-H functionalization. Additionally, these valuable alternative electrophile coupling partners could be used in ecofriendly solvent diethyl carbonate. Finally, these efficient coupling conditions were successfully extended to sulfonates such as tosylates and mesitylates. Further studies would address metalcatalyzed reactions promoting the more reluctant phenolate derivatives we identified herein such as carbonates and acetates.

Experimental Section

General procedure for ruthenium-catalysed N-directed ortho-C-H diarylation. (i) In trifluoromethylbenzene: as a typical experiment, in an oven-dried 20 ml Schlenk tube equipped with a magnetic stirring bar was charged phenyl-1H-pyrazole (66 l, 0.5 mmol), phenyl triflate (320 l, 1.5 mmol), K2CO3 (276 mg, 2 mmol), [RuCl2(p-Cym)]2 (7.8 mg, 2.5 mol%) and PivOH (15 mg, 30 mol%) in trifluoromethylbenzene (4 ml). The mixture was stirring at 110 °C under argon during 22 h. After extraction (CH2Cl2/H2O), the organic layer was removed in vacuum. The crude product was purified by column chromatography on silica or recrystallization (heptane/ethyl acetate) and affords the corresponding product. (ii) In diethyl carbonate (DEC): as a typical experiment, in an oven-dried 20 ml Schlenk tube equipped with a magnetic stirring bar was charged phenyl triflate (320 l, 1.5 mmol), KOAc (196 mg, 2 mmol) and [RuCl2(p-Cym)]2 (7.8 mg, 2.5 mol%) and placed under vacuum for 20 minutes. Under argon, phenyl-1H-pyrazole (66 l, 0.5 mmol) was added with diethyl carbonate (4 ml). The mixture was stirring at 110 °C under argon during 22 h. After extraction (CH2Cl2/H2O), the organic layer was removed in vacuum. The crude product was purified by column chromatography on silica or recrystallization (heptane/ethyl acetate) and affords the corresponding product.

  Conditions: phenyl-1H-pyrazole (1 equiv.), phenyl triflate (3 equiv.), base (4 equiv.), [RuCl2(p-Cym)]2 (2.5 mol%), additive (30 mol%), solvent [0.125 M based on phenyl-1H-pyrazole], 22 h, argon. Conversion and yields are determined by 1 H NMR based on phenyl-1H-pyrazole. PhCF3: trifluoromethylbenzene; Ac-Val-OH: N-Acetyl-L-valine;

Scheme 2 .

 2 Scheme 2. Phenolate derivatives in Ru-catalyzed ortho-sp 2 -C-H arylation of arylpyrazole.

Table 4 .

 4 Sulfonates and carbonates from 2-naphthol (20).

	entry	Reagent (equiv.)	Leaving group	Isolated Yield (%)

Acknowledgements

This work was by supported by the CNRS, Université de Bourgogne, Conseil Régional de Bourgogne through the plan d'Actions Régional pour l'Innovation (PARI, program CDEA, 3MIM) and the Fonds Européen de Développement Régional (FEDER). Thanks are due to S. Royer for some triflate reagents preparations.

Keywords: biphenyl • C-H activation • dichloroethane • sulfonates • triflates