Thomas Gauthier

Gabriel Vigny

DISTRIBUTION OF POINTS WITH PRESCRIBED DERIVATIVE IN POLYNOMIAL DYNAMICS by

In analogy to the equidistribution of preimages of a prescribed point by the iterates of a polynomial map f in C towards the equilibrium measure, we show here the equidistribution of points z for which (f n ) ′ (z) = a for suitable a towards the equilibrium measure. We then give a similar statement in the space of degree d polynomials for the equidistribution of parameters for which the n-derivative at a given critical value has a prescribed derivative towards the activity current of the corresponding critical point.

Introduction

In the first part of the article, we are interested in the equidistribution of points with prescribed derivative for a polynomial map f : C → C of degree d ≥ 2 in C. Recall for that that the Green function of f is

g f (z) := lim n→∞ d -n log max {1, |f n (z)|} , z ∈ C .
The Julia set J f of f is J f := ∂{g f = 0}. The probability measure µ f := dd c z g f is known as the equilibrium measure of f . It is the unique measure of maximal entropy log d of f and its support is the Julia set of f . This measure describes many equidistribution phenomena, notably the following: there exists a set E containing at most one point such that for all a ∈ C\E, the measure equidistributed on the preimages of a converges to µ f :

lim n 1 d n f n (z)=a δ z = µ f
in the sense of measure (we take into account the multiplicity in the sum). This result is due to Brolin [B] and has been extended to the case of rational maps by Ljubich [L] and independently by Freire, Lopes and Mañé [FLM], a quantified statement has been established in [DO]. See [START_REF] Erik | Complex dynamics in higher dimensions[END_REF][START_REF] Briend | Deux caractérisations de la mesure d'équilibre d'un endomorphisme de P k (C)[END_REF][START_REF] Dinh | Dynamique des applications d'allure polynomiale[END_REF] for similar results in higher dimensions.

We want to understand a similar statement but for derivatives. Let us give some motivations for that :

-the solutions of f n (z) = z are exactly the n-periodic points which are known to equidistribute towards the equilibrium measure (see [L]). Then, one can ask what is exactly the multiplicity of the solutions and for that one need to solve f n (z) = z and (f n ) ′ (z) = 1. Then, if the solutions of (f n ) ′ (z) = 1 were far from the Julia set, one could conclude that the solution of f n (z) = z are mostly simple. In fact, Theorem A below says that the solutions of (f n ) ′ (z) = 1 tends to accumulate on the Julia set J f of f , making that approach ineffective. Similarly, assume that one wants to compute the Lyapunov exponent of f by computing the derivative f n (z) at some generic point z in the support of µ f , Theorem A shows that a small error in the selection of z can give any possible result for the derivative! -by the chain rule, the map (z, n) → (f n ) ′ (z) defines a cocycle, the article thus addresses the question of the equidistribution of the preimages of a prescribed target by a cocyle in the simplest case. -more generally, we think of polynomial maps of C as a test case, the questions raised in the article can be asked in a lot of situations (e.g. Hénon mappings, rational maps in higher dimension). We will study the equidistribution towards activity currents in the second part of the article. We now state our results. For λ ∈ C, we denote by ν λ n the following probability measure:

ν λ n := 1 d n -1 (f n ) ′ (z)=λ δ z ,
where the sum is taken with multiplicity. Our first result is the following equidistribution statement of ν λ n towards the equilibrium measure µ f of f : Theorem A. -Let f : C -→ C be a polynomial map of degree d ≥ 2 and let ν λ n and µ f be the measures defined above. There exists a polar set E f ⊂ C * such that 1. For all λ ∈ C \ E f , one has ν λ n → µ f in the sense of measures, 2. If f has no siegel disk and no escaping critical points, one has

E f = ∅, 3. If f is hyperbolic, one also has then E f = ∅.
Recall that f is hyperbolic if it is uniformly strictly expanding on its Julia set; or equivalently if the set

{f n (c) ; n ≥ 0, f ′ (c) = 0} is disjoint from the Julia set J f of f .
The proof of the first point (see Theorem 2.6) is deduced from the following interesting proposition (see subsection 2.2 for the definition of PB measures):

Proposition 1.1. -Let f : C -→ C be a polynomial map of degree d ≥ 2 and let ν be a PB measure on P 1 , then we have the convergence:

lim n→∞ 1 d n -1 (f n ) ′ * (ν) = µ f Remark. - 1.
In particular, for quadratic polynomial maps f , the result is true for all λ = 0 as long as f does not have a Siegel disk. 2. The case λ = 0 is the equidistribution of the preimages of the critical sets which is known to fail if and only if some critical points is in the exceptional set, i.e. f is affine conjugate to z d . 3. Theorem A can be, at least partially, extended to rational maps of P 1 (see Remarks 2.4 and 2.6). Nevertheless, Theorem A is invariant under affine conjugacy whereas it is not under Moebius conjugacy hence we choose to stick to polynomial maps.

The idea of the proof of the first point of Theorem A is to study the dynamics of the tangent map

F (z, u) = (f (z), f ′ (z)•u) (in C 2 ).
We show that its Green current is in fact the pull back of µ f by the projection π 1 on the first coordinate and that (d n -1) -1 (F n ) * ([u = 1]) converges the Green current (Proposition 2.5). Then we show the convergence (for λ outside a pluripolar set) of the intersection

(d n -1) -1 (F n ) * ([u = 1]) ∧ [u = λ] towards π * 1 (µ f ) ∧ [u = λ]
which concludes the proof (Theorem 2.6). It should be possible to show that E f is empty with that approach using the recent theory of Dinh and Sibony of density of currents [START_REF] Dinh | Density of positive closed currents, a theory of non-generic intersections[END_REF], as we explain in Remark 2.4. We then give the proof of the second and third points of Theorem A using the classical approaches of Brolin and Ljubich, though they seem to generalize only to very specific other cases.

In the second part of the article, we focus on bifurcation phenomena in parameter spaces of polynomial maps of C of a given degree d ≥ 2. For c = (c 1 , . . . , c d-2 ) ∈ C d-2 and a ∈ C, we let

P c,a (z) := 1 d z d + d-1 j=2 (-1) d-j σ d-j (c) j z j + a d , z ∈ C ,
with σ k (c) the monic homogeneous degree k symmetric function in the c i 's. This family is known to be a finite branched cover of the moduli space P d of critically marked degree d polynomials, i.e. the space of affine conjugacy classes of degree d polynomials with d -1 marked critical points (see e.g. [START_REF] Dujardin | Distribution of rational maps with a preperiodic critical point[END_REF]§5]). The critical points of P c,a are exactly c 0 , . . . , c d-2 , with the convention c 0 := 0.

Pick 0 ≤ i ≤ d -2. As it now is classical, we say that a critical point c i is passive at (c * , a * ) ∈ C d-1 if there exists a neighborhood U ⊂ C d-1 of (c * , a * ) such that the sequence F i n : C d-1 → C of holomorphic maps defined by F i n (c, a) := P n c,a (c i
) is a normal family on U . Otherwise, we say that c i is active at (c * , a * ). The activity locus of c i is the set of parameters (c, a) ∈ C d-1 such that c i is active at (c, a). We can define an activity current T i to give a measurable sense to the notion of activity. We denote by π d-1 (resp. π 1 ) the canonical projections on C d-1 (resp. on C). Let T be the Green current of the map f (it can be defined as the limit of d -n (f n ) * (π * 1 (ω 1 ))). Then, by the work [BB1] of Bassanelli and Berteloot, we have T i := (π d-1 ) * (T ∧ [z = c i ]). The invariance of the Green current implies that it can also be defined by intersecting with the graph of the critical value P c,a (c i ) as below:

(1)

T i = 1 d (π d-1 ) * (T ∧ [z = P c,a (c i )]) .
As proved by Dujardin and Favre [DF], the current T i is exactly supported by the activity locus of c i . Moreover, they prove that the sequence of smooth forms d -n (F i n ) * ω P 1 converges in the weak sense of currents to T i .

The currents T i and i T i are known to equidistribute various phenomena: parameters for which the critical point c i is preperiodic with a given orbit portrait [DF], parameters admitting a cycle with a given multiplier [START_REF] Bassanelli | Distribution of polynomials with cycles of a given multiplier[END_REF][START_REF] Bassanelli | Lyapunov exponents, bifurcation currents and laminations in bifurcation loci[END_REF][START_REF] Buff | Quadratic polynomials, multipliers and equidistribution[END_REF][START_REF] Gauthier | Equidistribution towards the bifurcation current I: multipliers and degree d polynomials[END_REF], or parameters at which the critical points are sent to some prescribed target [D, GV].

As in the case of polynomial maps, we prove here the following:

Theorem B. -Pick any integer 0 ≤ i ≤ d -2.
Then the following convergences holds in the weak sense of currents on C d-1 : 1. for any probability measure ν with bounded potential on C,

lim n→∞ 1 d n C (P n c,a ) ′ (P c,a (c i )) = λ dν(λ) = T i ,
2. there exists a polar set

E i ⊂ C * such that, for any λ ∈ C \ E i , lim n→∞ 1 d n (P n c,a ) ′ (P c,a (c i )) = λ = T i .
Finally, we focus on the quadratic family p c (z) := z 2 + c, c ∈ C. In that very particular context, we can prove a stronger statement, which in particular imply the exceptional set is empty (see Theorem 3.7). Here this means that for any λ ∈ C, the sequence of finite measures 1

2 n [(p n c ) ′ (c) = λ] converges to the harmonic measure of the Mandelbrot set.

2. In the phase space of a complex polynomial

The aim of the present section is to prove Theorem A.

A tangent map

Let f be a polynomial map of C of degree d ≥ 2. We can write it as f (z) = d i=0 a i z i . We consider the tangent map F (z, u) = (f (z), f ′ (z) • u) acting on the tangent bundle Tan(C) that we write in the birational model P 1 × P 1 as:

F ([z : t], [u : v]) = d i=0 a i z i t d-i : t d , d i=0 ia i z i-1 t d-i u : t d v .
If c 1 , . . . , c d-1 are the critical points of f in C (counted with multiplicity), then the indeterminacy points of F are the points I j = (c j , [1 : 0]) and the points I ∞ := ([1 : 0], [0 : 1]). We denote by I(F ) the union of those points. Observe that the (invariant

) set {[1 : 0]}×P 1 \I ∞ is sent to I ′ := ([1 : 0], [1 : 0]
) by F . Finally, observe that the point I ′ is attracting.

Let ω i be the pull back of the Fubini-Study form ω P 1 on P 1 by the canonical projection to the i-th factor of π i :

P 1 × P 1 → P 1 . Let {ω i } denote the class of ω i in H 1,1 (P 1 × P 1 ). Lemma 2.1. -The map F is algebraically stable i.e. (F * ) n = (F n ) * in H 1,1 (P 1 × P 1 ).
The action of F * is given in the ({ω 1 }, {ω 2 }) basis by the matrix:

d d -1 0 1 .
Its topological degree d t is d, in particular, it is not cohomologically hyperbolic.

Proof. -The obstruction to the algebraic stability is the fact that some hypersurface (i.e. some curve here) is sent to some indeterminacy point by some iterate F n of F ( [S]). As f is holomorphic, the only possibility is that some {p} × P 1 is sent to an indeterminacy set by F n . By the chain rule, F n is the extension of

F n (z, u) = (f n (z), (f n ) ′ (z) • u) on the tangent bundle Tan(C) to its compactification P 1 × P 1 . Let p ∈ C, observe that {p} × {[0 : 1]} ⊂ {p} × P 1 is sent to {f (p)} × {[0 : 1]} whether p is critical or not.
It follows from this that no hypersurface {p} × P 1 is sent to an indeterminacy point by F n for p ∈ C.

On the other hand, the point

I ′ is sent to itself by F so F n ([1 : 0] × P 1 ) = I ∞ . So F is algebraically stable.
The rest of the proof follows. Observe that the first dynamical degree is by definition the spectral radius of the matrix

d d -1 0 1 hence it is equal to d = d t , by definition, it implies that F is not cohomologically hyper- bolic.
The study of cohomologically hyperbolic maps is very well developed (see e.g. [Gu]), no general theory exists for not cohomologically hyperbolic.

Basics on dsh functions and PB measures

Let us recall some facts on dsh functions (see e.g. [START_REF] Dinh | Dynamics of regular birational maps in P k[END_REF]). Recall that a probability ν in (P 1 ) 2 has bounded quasi-potentials (or is PB) if ν admits a negative quasi-potential U (dd c U + Ω = ν where Ω is some smooth probability measure) such that | U, S | ≤ C for any positive smooth form S of bidegree (1, 1) and mass 1. Such notion of quasi-potential can be extended to any positive closed current of bidegree (1, 1) and mass 1 with the same bound | U, S | ≤ C. In particular, any smooth measure is PB.

Definition 2.2. -We say that a function ϕ on (P 1 ) 2 is dsh if, outside a pluripolar set, it can be written as a difference of quasi-psh functions.

For example, if ϕ ∈ C 2 , then it is dsh. Let DSH P 1 2 be the space of such functions on (P 1 ) 2 . For any ϕ ∈ DSH P 1 2 , we write dd c ϕ = T + -T -where T ± are positive closed currents of bidegree (1, 1). Let ν be a PB measure on (P 1 ) 2 . The following defines a norm on the space DSH P 1 2 :

ϕ ν := ϕ L 1 (ν) + inf T ± ,
where the infimum is taken on all the decompositions dd c ϕ = T + -T -as above. It turns out that taking another PB measure ν ′ gives an equivalent norm on DSH P 1 2 (see e.g. [START_REF] Dinh | Dynamics in several complex variables: endomorphisms of projective spaces and polynomial-like mappings[END_REF]p. 283]).

Finally, recall that a Borel set that is of measure 0 for all the PB measures is in fact pluripolar.

A first convergence property

Recall that the Green measure µ f of the map f can be defined as the limit

µ f := lim n d -n (f n ) * (ω P 1 ) .
It then follows from the fact that F is a skew-product that the sequence of positive closed current d -n (F n ) * (ω 1 ) converges to the Green current T F of F which is a positive closed current of mass 1 and that

T F = π * 1 (µ f ). Lemma 2.3. -The sequence of positive closed currents d -n (F n ) * (ω 2 ) converges to the Green current T F of F . Proof. -Write ω 2 = ω 1 + ω 2 -ω 1 . Since d -n (F n ) * (ω 1 ) converges to T F , all there is left to prove is that d -n (F n ) * (ω 2 -ω 1 ) converges to 0 in the sense of currents. Observe that F * (ω 2 -ω 1 ) is cohomologous to ω 2 -ω 1 hence we can write F * (ω 2 -ω 1 ) = ω 2 -ω 1 + dd c ϕ
where ϕ is smooth outside I(F ) and is a dsh function (in fact ϕ is quasi-psh since F * (ω 1 ) is a smooth form).

Let W be a small neighborhood of I ′ such that F (W ) ⊂ W and where ϕ is uniformly bounded. Let ν W be a smooth (hence PB) probability measure with support in W . Then, a straight-forward induction gives:

d -n (F n ) * (ω 2 -ω 1 ) = d -n (ω 2 -ω 1 ) + dd c d -n n-1 k=0 ϕ • F k . The sequence of function ϕ n := d -n n-1 k=0 ϕ • F k is then a sequence of dsh functions. Furthermore, as F (W ) ⊂ W , we see that ϕ n ∞,W ≤ nd -n ϕ ∞,W ≤ C
where C is a constant that does not depend on n. In particular, ϕ n ν W is uniformly bounded. As all the DSH-norms are equivalent, we deduce that (ϕ n ) is bounded in L 1 for the standard Fubini Study measure in (P 1 ) 2 .

On the other hand, the sequence d -n (F n ) * (ω 2 ) is bounded in mass hence we can extract a converging subsequence. Its limit is a positive closed current cohomologuous to ω 1 hence it can be written as π * (ν ′ ) where ν ′ is some probability measure in P 1 . Extracting again, we can assume that (ϕ n ) converges in L 1 (it is bounded in DSH). Its limit V satisfies dd c V = (π 1 ) * (ν ′µ f ) hence it is constant on each fiber of π 1 . In other words, V = π * 1 (v) for some dsh function v on P 1 . Take z ∈ C in the interior of the filled Julia set. Then, the sequences (f n (z)) and (f n ) ′ (z) are equicontinuous near z, hence bounded. In particular, for any u ∈ C, we have that the sequence (f n (z), (f n ) ′ (z) • u) stays in a some compact subset of C × C (this compact set can even be chosen to be uniform in a neighborhood of (z, u)). In particular, the sequence ϕ n (z, u) converges to 0. Similarly, for [z : t] in the basin of attraction of ∞ and any [u : v] with u = 0, we have that the sequence ([f n (z : t) :

t d n ], [(f n ) ′ (z : t)u : t d n -1 v]) converges to ([1 : 0], [1 : 0]). Hence, ϕ n ([z : t], [u : v]) converges to 0 for such ([z : t], [u : v]).
It follows that the function v is equal to 0 in the Fatou set. Now, the Julia set has empty interior and a dsh function that is 0 outside such set is identically 0 by pluri-fine continuity. It follows that ν ′ = µ f which ends the proof.

Any PB probability measure ν in P 1 can be written as ω P 1 + dd c η where η is bounded in P 1 so we have the following:

Corollary 2.4. -For any PB probability measure ν in P 1 , the sequence of currents d -n (F n ) * (π * 2 (ν)) converges to T F in the sense of currents.

The convergence Theorem

For u 0 ∈ C, we denote by [u = u 0 ] the current of integration on the line P 1 × {u 0 } of P 1 × P 1 .

Proposition 2.5. -For any u 0 ∈ C * , the sequence of currents d -n (F n ) * [u = u 0 ] converges to T F in the sense of currents.

Proof. -By the above Corollary 2.4, one deduces that outside a polar set of We can now prove the first part of Theorem A:

u 0 ∈ C, d -n (F n ) * ([u = u 0 ]) converges to T F in the sense of currents. Choose such a u 0 = 0. For λ ∈ C * , consider the map D λ : (z, u) → (z, λu). Observe that d -n (F n ) * ([u = 1 λ u 0 ]) = D * λ (d -n (F n ) * ([u = u 0 ])). Now, D * λ (d -n (F n ) * ([u = u 0 ])) converges to D * λ (T F )
Theorem 2.6. -Let ν be a probability PB measure in C, then

lim n→∞ 1 d n ((f n ) ′ ) * ν = µ f ,
weakly on C. In particular, outside a polar set of λ ∈ C, we have lim

n→∞ ν λ n = µ f . Proof. -Let u 0 ∈ C * , then observe that (F n ) * ([u = u 0 ]) ∧ [u = 1] = (F n ) * ([u = 1]) ∧ [u = u -1 0 ]
. Let inv : P 1 → P 1 be the rational map z → z -1 . In other words, (2)

(F n ) * ([u = u 0 ]) ∧ [u = 1] = (F n ) * ([u = 1]) ∧ π * 2 (inv * ([u = u 0 ]))
. By Fubini, we have for ν

(F n ) * (π * 2 (ν)) ∧ [u = 1] = (F n ) * ([u = 1]) ∧ π * 2 (inv * (ν)
). As remarked earlier, proving the statement for one specific PB measure ν implies the result for all PB measures. In particular, we can take ν = ω P 1 . By the above Proposition 2.5,

d -n (F n ) * ([u = 1]) → T F in the sense of currents, hence d -n (F n ) * ([u = u 0 ]) ∧ π * 2 (inv * (ω P 1 )) → T F ∧ π * 2 (inv * (ω P 1 )) since π * 2 (inv * (ω P 1 )
) has continuous potentials. By continuity of (π 1 ) * , we deduce that

(π 1 ) * d -n (F n ) * ([u = u 0 ]) ∧ π * 2 (inv * (ν)) → (π 1 ) * (T F ∧ π * 2 (inv * (ν))) = µ f
, in the sense of currents. Using (2) gives the first part of the theorem.

The second part follows since the set of λ for which

lim n→∞ 1 d n (f n ) ′ (z)=λ δ z = µ f
does not hold has zero measure for all the PB measure of C. Such a set is polar.

Remark. -1. Such a proof has the advantage to be both intuitive and geometric. It may easily be adapted to rational maps of P 1 (some new arguments are needed for maps whose Julia set is the whole P 1 ) or, as we will see in the next section, to parameter spaces of any dimension. 2. Nevertheless, it does not give the equidistribution for all λ but 0 without further arguments. This should be possible using the recent theory of Dinh and Sibony of density of currents [START_REF] Dinh | Density of positive closed currents, a theory of non-generic intersections[END_REF]. Indeed, let G(Tan(C), 1) be the Grassmanian of 1plan in Tan(C) and let us work in the birational model (P 1 ) 3 . We consider the coordinates ([z : t], [u : v], [a : b]). For an hypersurface A in (P 1 ) 2 , we let A denote the incidence variety associated to A in (P 1 ) 3 . In particular (working in the chart C 2 ), for (F n ) * {u = λ}, we have the cartesian equation of (F n ) * {u = λ}:

(f n ) ′ (z)u = λ and (f n ) ′′ (z)ua + (f n ) ′ (z)b = 0. Then observe that [ (F n ) * {u = λ}] is cohomologous (or at least bounded in cohomol- ogy) to: 2(d n -1)ω 1 ∧ ω 2 + (d n -1)ω 1 ∧ ω 3 + ω 2 ∧ ω 3 (for the coefficient before ω 1 ∧ ω 2 , choose a generic [a 0 : b 0 ] and count the number of solutions of (f n ) ′ (z)u = λ, (f n ) ′′ (z)ua 0 + (f n ) ′ (z)b 0 = 0, for that, replace u by λ((f n ) ′ (z)) -1
in the second equation). In particular, consider the sequence of positive closed currents (of bidegree (2, 2)) : 1 d n (F n ) * {u = λ} . We can extract a subsequence which converges to a limit T . Then, proceeding as in [START_REF] Dinh | Equidistribution of saddle periodic points for hénontype automorphisms of C k[END_REF], the result would follow for λ provided that:

T [u = 1] = 0
where [u = 1] is the line given by u = 1, b = 0 in (P 1 ) 3 . We were unable to prove this though heuristc arguments show that T should be T

F ∧ [a = 0] + 2π * 1 (µ f ) ∧ [u = 0].

Complement 1: Brolin's approach

In the rest of the section, we want to present two distinct alternative proofs, giving the second and third items of Theorem A. The first one is an adaptation of the classical proof of Brolin concerning the distribution of preimages. The second one is an adaptation of the proof of Lyubich concerning the distribution of preimages, which consists in building sufficiently many inverse branches.

In this subsection, we assume that f has no Siegel disk. Up to conjugating by a linear map, we can assume that f is unitary. In order to simplify the arguments, we shall also assume that no critical points are in the basin of attraction of infinity (this assumption is only technical). Let λ ∈ C * , recall that we denoted:

ν λ n := 1 d n -1 (f n ) ′ (z)=λ δ z
counting the multiplicity in the sum. We shall prove in this section:

Theorem 2.7. -Let f be a degree d ≥ 2 polynomial satisfying the above properties and pick any λ ∈ C * . Then the sequence (ν λ n ) n converges towards µ f in the sense of measures.

Observe that (ν λ n ) is a sequence of probability measures so we can extract a converging subsequence towards a limit µ ′ . We shall show that µ ′ = µ f . Lemma 2.8. -The support of µ ′ is contained in the Julia set J f of f .

Proof. -Let ε > 0 and consider J ε an ε-neighborhood of J f . Let U be a bounded Fatou component of f . Then in U \ J ε , the sequence (f n ) is normal and converges uniformly towards a constant, since f has no Siegel disk. So the sequence ((f n ) ′ ) is also normal and converges uniformly towards 0. In particular, it does not take the value λ in U ∩ J c ε for n large enough. Finally, assume U is the basin of attraction of ∞. As no critical point of f lies in U , then the same result follows from the fact that (

|(f n ) ′ |) converges uniformly to ∞ in U \ J ε .
We now come to the proof of Theorem2.7.

Proof of Theorem 2.7. -Consider the logarithmic potential u n of ν λ n defined by :

u n := 1 d n -1 log |(f n ) ′ -λ| - 1 d n -1 log d n
where the additive constant 1 d n -1 log d n is chosen to take into account the fact that (f n ) ′ is not unitary since the coefficient of its dominating term is d n . We want to apply [START_REF] Brolin | Invariant sets under iteration of rational functions[END_REF]Lemma 15.5] which, in our case, say that if the limit inferior of the u n is non-positive on J f then µ ′ = µ f , ending the proof. Observe that |f ′ | is uniformly bounded in the compact set J f (say by a constant M ) so that |(f n ) ′ | ≤ M n by the chain rule, since J f is a f -invariant set. In particular:

log |(f n ) ′ -λ| ≤ log(|(f n ) ′ | + |λ|) ≤ log(M n + |λ|) on J f . Hence, the logarithmic potential u n of ν λ n satisfies u n ≤ (d -n log(M n + |λ|) - 1 d n -1 log d n on J f ,
which goes to 0 with n. The result follows.

Remark. -1. Using potential theory is very efficient. Observe that we can prove, using the same proof, the equidistribution of the measures ν λ,k n defined by:

ν λ,k n := 1 d n -k (f n ) (k) (z)-λ δ z
for any λ = 0. 2. Nevertheless, the method works only for polynomials (Brolin method already fails for rational maps in the case of the equidistribution of preimages of a point). Moreover, it can be easily adapted to the setting of bifurcations only in special cases, as the quadratic family (see e.g. §3.4).

Complement 2: Lyubich's Inverse branches approach

In this section, we prove the following theorem:

Theorem 2.9. -Assume that f is a hyperbolic polynomial of degree d ≥ 2. Then, for all λ = 0, we have the equidistribution:

lim n→∞ ν λ n = µ f ,
in the sense of measures.

Proof. -We can assume that no critical point is in the exceptional set : the preimages of any critical point accumulate toward the equilibrium measure (if not f = z d and the result follows from easy direct computations). We first assume that no critical points is sent to another after some iterates and then explain what modifications need to be done to get to the general case. In particular, no critical point is periodic. With our assumptions, the Julia set is uniformly expanding: choosing some smooth metric in a neighborhood J η of J f , there exists ρ > 1 such that the image of a ball of radius δ in J η contains a ball of radius ρδ.

); -for (c, i, m) = (c ′ , i ′ , m ′ ), we have B i,m c ∩ B i ′ ,m ′ c ′
= ∅ (this is clear for m small enough by restricting the B c and it follows from the expansivity when all the preimages are in J η ).

Let a := sup x∈∂{∪ k-1 j=0 f -j (Bc)} |(f k ) ′ (x)| > 0 (no critical point lies in that set) and choose M so that M • a ≥ 2|λ|. For m ′ large enough, we have that for all z ∈ B i,m c , |(f n ) ′ (z)| ≥ M
for all m > n ≥ m ′ (this follows from the expansivity of J η ). Take m ≥ m ′ + k and let m > n ≥ mk. Then for x ∈ ∂B i,n c , we have by the chain rule and the above:

|(f m ) ′ (x)| = |(f n ) ′ (x)| • |(f m-n ) ′ (f n (x))| ≥ M • a ≥ 2|λ| .
On the other hand, if c i,n denotes the preimage by f n of c that belongs to B i,n c , then:

(f m ) ′ (c i,n ) = 0.
By the mean value theorem, we deduce that there exists a point x c,i,n ∈ B i,n c such that (f m ) ′ (x c,i,n ) = 0. Summing over all i times the number of critical points gives (d -1)d n such points. Summing over all n gives d md m-k such points. In particular, we have the decomposition:

ν λ n = 1 d m -1 c m-k≤n<m i δ x c,i,n + θ n ,
where θ n is a measure of mass d m-k d m -1 which can be taken < ε for k large enough. Denote:

1 d n i δ x c,i,n := µ c,n .
Then µ c,n is a probability measure which is known to converge to µ when n → ∞. Indeed, the sequence of measures equidistributed on the preimages of c converges to µ when m → ∞ and |x c,i,nc i,n | → 0 uniformly in i. Combining the above we have:

ν λ n = 1 d m -1 c m-k≤n<m d n d m -1 µ c,n + θ n → µ f ,
in the sense of measures.

Let us now explain what are the modifications in the case where one critical point is sent to another after some iterates (assume that all the critical points are simple): given m 0 large enough, using Lyubich inverse branches ideas (see e.g. [L, BD]) we can construct (1ε)d m 0 preimages of some small disks centered around each of the critical points. Removing some of those preimages, we can assume that (1 -2ε)d m 0 of those preimages lie in the expanding neighborhood J η of J f . However for two of such preimages, B and B ′ may satisfy f j (B) ∩ B ′ = 0, this will happen if and only if f j (c) = c ′ where c and c ′ are the critical points whose preimages by f j are in B and B ′ . Whenever that happen, we remove B ′ from the list of preimages that we keep. Now, we can construct for each of those disks exactly d m preimages by f m for all m (in particular, we take m ≫ j for all j as above). Now, when counting as above the points x in the preimages (more precisely their image by f l for k ≥ l > 0) such that (f m ) ′ (x) = 0, observe that the preimages such that f j (B) ∩ B ′ give several such points x in their preimages because of the multiplicity of (f m ) ′ (c i,n ) = 0 (c i,n is the preimage of the critical point c which is also a preimage of the point c ′ ). The rest of the proof is similar.

Finally, if some critical points has multiplicity, the argument is the same taking into account that in each preimages, one has to take several x counting the multiplicity.

Remark. -1. The proof extends easily to the case of rational maps, with the same hypothesis on the critical points and hyperbolicity of the Julia set. It has also the interest to show what is the "typical" behavior of a point x such that (f n ) ′ (x) = λ): the orbits stay near the Julia set for a long time, "gaining" hyperbolicity, then it is "ejected" and goes very close to a critical point, losing that hyperbolicity then its orbit follows the orbit of the critical point for a small time. Notice that the proof could be improved to work with weakened hypothesis using more subtlety inverse branches ideas and Pesin theory. 2. Nevertheless, the method of inverse branch cannot be used in the setting of bifurcations (or Hénon maps).

In the parameter space of polynomial maps

We consider now the following polynomial map in C d-1 × C defined in the introduction :

f : C d-1 × C -→ C d-1 × C ((c, a), z) -→ ((c, a), P c,a (z)).
Our aim in this section is to prove Theorem B.

A partial tangent map

We consider the "partial" tangent map:

f : C d-1 × C × C -→ C d-1 × C × C ((c, a), z, u) -→ ((c, a), P c,a (z), P ′ c,a (z) • u)
We shall still denote by f its homogenous extension to P d+1 (C) × P 1 (C):

f : P(C) d × P 1 (C) -→ P(C) d × P 1 (C) ([c : a : z : t], [u : v]) -→     ct d-1 : at d-1 : 1 d z d + d-1 j=2 (-1) d-j σ d-j (c) j z j + a d : t d   ,   z d-1 + d-1 j=2 (-1) d-j σ d-j (c)z j-1 )u : t d-1 v     .
In homogenous coordinate, with the convention c 0 = 0, the indeterminacy set of f is

I( f ) ={t = 0} ∩    1 d z d + d-1 j=2 (-1) d-j σ d-j (c) j z j + a d = 0    ∪ {t = 0} ∩ {u = 0} ∪ d-2 i=0 {z = c i } ∩ {v = 0}.
Observe that the (invariant) set {[c : a : z : 0]} × P 1 \I( f ) is sent by f to the fixed point I ′ := ([0 : 0 : 1 : 0], [1 : 0]). For the rest of the text, we let W be a small neighborhood of I ′ such that f (W ) ⊂ W . Observe that one cannot have f (W ) ⋐ W because the fixed point I ′ has neutral directions given by the parameters space variables.

Let ω 1 (resp. ω 2 ) be the pull back of the (normalized) Fubini-Study form ω P d of P d (resp. ω P 1 on P 1 ) by the canonical projection to the first (resp. second) factor P 1 : P d × P 1 → P d (resp. P 2 : P d × P 1 → P 1 ). Let {ω i } denote the class of ω i in H 1,1 (P d × P 1 ).

Lemma 3.1. -The map f is algebraically stable i.e. ( f * ) n = ( f n ) * in H 1,1 (P d × P 1 ). The action of ( f ) * is given in the ({ω 1 }, {ω 2 }) basis by the matrix:

d d -1 0 1 .
Proof. -We again rely on the characterization of the algebraic stability by the existence of some hypersurface sent to the indeterminacy set by some iterate (see [S]). Let A be such an hypersurface, then P 1 (A) is an algebraic subvariety of P d . Assume P 1 (A) = P d . Let [c : a : z : t] be a fixed point of f such that f ′ (z) = 0 (it is obvious that there are infinitely many such points), and let [u : v] ∈ P 1 such that ([c : a :

z : t], [u : v]) ∈ A. By the chain rule, f n ([c : a : z : t], [u : v]) = ([c : a : z : t], (f n ) ′ (z) • u) / ∈ I( f )
. This is a contradiction.

In particular, P 1 (A) = P d so that A = H × P 1 . We extend f as a rational mapping to P d by its homogenous extension to P d and still denote it by f . Observe first that the map f is algebraically stable: indeed, if it is not the case, since f is a polynomial map in C d , the only possibility is that {t = 0} is sent to the indeterminacy set. This is not the case since f

({t = 0})\I(f ) = [0 : 1 : 0]. If H = {t = 0}, let p ∈ H ∩ C d . By the chain rule, f n is the extension of f n (p, u) = (f n (p), (f n ) ′ (p) • u). Observe that {p} × {[0 : 1]} ⊂ {p} × P 1 is sent to {f (p)} × {[0 : 1]}
whether p is critical or not. It follows from this that the line {p} × P 1 is not sent to the indeterminacy set by f n . We assume now that H = {t = 0}. As the point I ′ is sent to itself by f then f n (I ′ ) = I ′ / ∈ I( f ). So f is algebraically stable. The rest of the proof follows.

Convergence towards the Green current of the tangent map

In the sequel, we let T be the Green current of the map f . One result which follows applying the same proof as that of Lemma 2.3is the following.

Lemma 3.2. -The sequence of positive closed currents d -n ( f n ) * (ω 2 ) converges to the Green current T f = P * 1 (T ) of f .

Unfortunately, this result itself is not useful in our case. Indeed, to define the activity current T i one needs to intersect the Green current T f with the current of integration on {z = P c,a (c i )}. Such an operation is not continuous in the sense of currents, hence Lemma 3.2 does not provide sufficient informations for our purpose. Instead, we shall prove directly the following:

Proposition 3.3. -The sequence (d -n ( f n ) * (ω 2 ) ∧ [z = P c,a (c i )]
) n of positive closed currents of bidegree (2, 2) is well defined and converges to the current

T f := P * 1 (T ) ∧ [z = P c,a (c i )] .
Proof. -Observe first that I( f n ) ∩ {z = P c,a (c i )} has codimension 3 (this follows from the fact that {z

= P c,a (c i )} ∩ {P n-1 c,a (z) = c i } has codimension 2 in C d × C. In particular, d -n ( f n ) * (ω 2 )∧[z = P c,a (c i )] is well defined since the restriction of the current d -n ( f n ) * (ω 2 ) to [z = P c,a (c i )] is smooth outside I( f n ). Write ω 2 = ω 1 + ω 2 -ω 1 . Since d -n ( f n ) * (ω 1
) converges to T f with local uniform convergence of the potentials, all there is left to prove is that d

-n ( f n ) * (ω 2 -ω 1 ) ∧ [z = P c,a (c i )] converges to 0 in the sense of currents. Observe that f * (ω 2 -ω 1 ) is cohomologous to ω 2 -ω 1 hence we can write f * (ω 2 -ω 1 ) = ω 2 -ω 1 + dd c ϕ
where ϕ is smooth outside I( f ) and is a dsh function. A straight-forward induction gives:

d -n ( f n ) * (ω 2 -ω 1 ) = d -n (ω 2 -ω 1 ) + dd c d -n n-1 k=0 ϕ • f k .
We write

ϕ n := d -n n-1 k=0 ϕ• f k . We have reduced the problem of proving the convergence of d -n ( f n ) * (ω 2 ) ∧ [z = P c,a (c i )] to T f = P * 1 (T ) ∧ [z = P c,a (c i )]
to proving that the sequence (ϕ n ) restricted to {z = P c,a (c i )} converges to 0 in L 1 ({z = P c,a (c i )}). We denote by φn the restriction of ϕ n to {z = P c,a (c i )}.

The sequence of functions ( φn ) is then a sequence of dsh functions on {z = P c,a (c i )} by construction. Furthermore, if U denotes a stability component of c i on which its orbit is bounded, then for all V ⋐ U , the families (P n c,a (c i )) and ((P n c,a ) ′ (c i )) are equicontinuous hence equibounded. In particular, for any compact set K ′ ∈ C, there exist compact sets L ⊂ C and K ⊂ C such that for all (c, a) ∈ V and all u ∈ K ′ , for any n ∈ N * , one has:

P n c,a (P c,a (c i )) ∈ L and ((P n c,a ) ′ (P c,a (c i )) • u) ∈ K ′ . Let ν L×K be a smooth (hence PB) probability measure with support in L × K. Since ϕ is smooth in C d × C, we have φn ∞,V ≤ nd -n ϕ ∞,L×K ≤ C where C is a constant that does not depend on n. In particular, ϕ n ν L×K is uniformly bounded. As all the DSH-norms are equivalent, we deduce that ( φn ) is bounded in L 1 for the standard Fubini Study measure in {z = P c,a (c i )}.

On the other hand, the sequence d -n ( f n ) * (ω 2 ) ∧ [z = P c,a (c i )] is bounded in mass hence we can extract a converging subsequence. Its limit is a positive closed current cohomologuous to ω 1 ∧ [z = P c,a (c i )] hence it can be written as π * (T ′ ) where T ′ is some (1, 1) current in [z = P c,a (c i )] and π denote the projection to the first coordinate of {z = P c,a (c i )}. Extracting again, we can assume that ( φn ) converges in L 1 (it is bounded in DSH ). Its limit V satisfies dd c V = π * (T ′ -T ) hence it is constant on each fiber of π. In other words, V = π * 1 (v) for some dsh function v on {(c, a, z) ; z = P c,a (c i )}. The above argument shows that v = 0 on any stability component of c i on which its orbit is bounded. The case of the component on which c i goes to ∞ is similar using this time a arbitrary compact sets of the form L × K where K is a compact set on the complementary of 0 ∈ C (indeed, the derivative converges to ∞).

It follows that the function v is equal to 0 in the stability locus. Now, the unstability locus has empty interior (this is classical and follows directly from Montel theorem) and a dsh function that is 0 outside such set is identically 0 by pluri-fine continuity. It follows that π * (T ′ ) = π * (T ) which ends the proof.

The following is now obvious: Corollary 3.4. -For any PB probability measure ν on P 1 , the sequence of currents d -n ( f n ) * (π * 2 (ν)) ∧ [z = P c,a (c i )] converges to T f in the sense of currents.

Value distribution of the derivative

For u 0 ∈ C, recall that we denote by [u = u 0 ] the current of integration on the hyperplane P d × {u 0 }. The proof we implement here can not be easily generalized to higher degrees to prove that exceptional sets from Theorem B are empty. Indeed, the pointwise estimates in C\M used in the proof we need have to be replaced with more elaborate estimates, as in [GV]. The estimates we can prove in that context concern only the fastest escaping critical point.

Moreover, this strategy is specifically designed for proving codimension 1 equidistribution phenomena, since the proof gives the L 1 loc convergence of the potentials. On the other hand, the proof of Theorem B we give above can be generalized to higher codimension objects.

  in the sense of currents by assumption and continuity of D * λ . Since T F = π * 1 (ν), we have D * λ (T F ) = T F . The result follows since taking a u 1 ∈ C * arbitrary, we can write λu 1 = u 0 for a suitable λ.

  Fix ε > 0 and k ∈ N large enough. Take some small ball B c around each critical points c, we consider the preimages f -m (B c ) for all c and all m ∈ N. Choosing B c small enough guarantees that each f -m (B c ) consists of d m distinct connected components that we denote B i,m c such that: -the diameters of B i,m c goes to 0 when m → ∞ by expansivity ; -for m ≥ m 0 , B i,m c ⊂ J η (all the preimages of the critical points end up in J η

  In particular, we get lim n→∞ 2-n log |(p n c ) ′ (c)| = g M (c) and (p n c ) ′ (c) → ∞ as n → ∞. To conclude, we just have to remark that lim c ) ′ (c) = g M (c) ,which concludes the proof of the Theorem.

Both authors are partially supported by the ANR grant Lambda ANR-13-BS01-0002. The first author is partially supported by a PEPS "Jeune-s Chercheur-e-s" Grant.

Proposition 3.5. -For any u 0 ∈ C * , the sequence of currents d -n ( f n ) * ([u = u 0 ]) converges to T f ∧ [z = P c,a (c i )] in the sense of currents.

Proof. -By the above Corollary 3.4, one deduce that outside a polar set of u 0 ∈ C, d -n ( f n ) * ([u = u 0 ]) ∧ [z = P c,a (c i )] converges to T f in the sense of currents. Choose such a u 0 = 0. For λ ∈ C * , consider the map D λ : (c, a, z, u) → (c, a, z, λu). Observe that

) in the sense of currents by assumption and continuity of

. The result follows since taking a u 1 ∈ C * arbitrary, we can write λu 1 = u 0 for a suitable λ.

We can now prove Theorem B. Let us restate it: Theorem 3.6. -Let ν be a PB measure in C, then we have the equidistribution:

In particular, outside a polar set of λ ∈ C, we have:

Proof. -Denote by p : C d → C d-1 the projection onto the parameter variables, i.e. the map defined by p(c, a, z)

)) are equal. By Fubini Theorem, we have for ν,

As before, it is enough to prove the first point for any PB measure, so we take ν = ω 2 the Fubini-Study form (inv * (ω 2 ) = ω 2 ). By the above proposition

) has continuous potentials. By continuity of (P 1 ) * and p * , and using the equality between the currents

gives the first part of the theorem. The second part follows since the set of λ for which the convergence

does not hold has zero measure for all the PB measures of C. Such a set is polar.

In the quadratic family

We now focus on the quadratic family parametrized by p c (z) := z 2 + c, for c ∈ C. We denote by K c the filled-in Julia set of p c and by J c the Julia set:

The bifurcation measure of the quadratic family is µ bif = 1 2 dd c g M . Its support is the boundary of the Mandelbrot set M . Moreover, the probability measure µ M = 2µ bif is the harmonic measure of the Mandelbrot set (see e.g. [CG]).

Pick a polynomial λ(c) ∈ C[c] and for any integer n ≥ 1, let

where we take into account the multiplicity in the sum. Beware that given a polynomial λ, the measure ν λ n has mass (2 n -1)/2 n ∼ 1 for n large enough. We want here to prove the following more general result.

Theorem 3.7. -For any polynomial λ(c) ∈ C[c], the sequence of finite measures (ν λ n ) n≥1 converges weakly to the harmonic measure µ M of the Mandelbrot set.

In [START_REF] Buff | Quadratic polynomials, multipliers and equidistribution[END_REF]Lemma 2], Buff and the first author prove the following we rely on:

This lemma is similar to the extremality property used in Brolin's approach in Section 2.5. . By definition, the sequence of subharmonic functions (u λ n ) is locally uniformly bounded from above. Hence, by Hartogs lemma, -either u λ n → -∞ uniformly locally on C, -or it admits subsequences which converge in L 1 loc (C). According to Lemma 3.8, to conclude it is sufficient to prove that u λ n → g M pointwise on C \ M . Indeed, if this holds true any L 1 loc limit u of (u λ n ) is subharmonic on C and coincides with g M on C \ M . By Lemma 3.8, this means that u λ n → g M in L 1 loc and the conclusion follows.

Pick now c ∈ C \ M , then there exists C > 0 universal such that


Guilaume Hawing

Patrice M Okouma
email: okouma@saao.ac.za

A New and Deterministic Scheme for Characterizing The Organization of Prime Numbers

Keywords: Number Theory, Composite Odd Numbers, Prime Numbers

, where we propose a new perspective on the Riemann hypothesis.

I. INTRODUCTION

Some numbers have the special property that they cannot be expressed as the product of two or more smaller numbers, e.g. 2, 3, 5, 7, etc. Such numbers are called prime numbers, and they play an important role, both in pure mathematics and its applications. Prime numbers occur in a very irregular way within the sequences of natural numbers. In particular, the distribution of prime numbers exhibits a local irregularity but a global regularity. One of the best existing results illustrating that global regularity is the prime number theorem giving the number of prime number, π(x), not exceeding an upper limit x. In epoch-marking works published in the late nineteenth century, it was shown that π(x) behaves asymptotically as x ln x [START_REF] Hadamard | [END_REF][START_REF] De La La Vallée-Poussin | Article in a journal[END_REF]. Prime numbers derive most of their peculiar importance from the fundamental theorem of arithmetic stating that any composite natural integer can be expressed in one and only one way as a product of prime number [START_REF] Gauss | Disquisitiones Arithmeticae[END_REF]. Given the importance of primes in the construction of all natural numbers, a problem that presents itself at the very roots of mathematics is therefore the organization of the primes among the integers.Formulated in other terms, a fundamental tool for number theory would be a scheme allowing to obtain, at a relatively low computational cost -when compared to existing approaches, the complete list of primes below any upper limit n. In the present work (Paper I), we propose a simple and self-consistent deterministic scheme allowing us to investigate further the intrinsic organization of prime numbers. From the said scheme, we have established an algorithm that systematically yields the complete list of prime numbers below any preassigned limit x at a relatively low computational cost. The reader might wish to test our scheme online for values of x ≤ 10 9 , on this website http://univgandhi-guinee.com/algo/public/home/nombres-premiers 1 . In Paper II ( [START_REF] Hawing | Riemann Hypothesis Revisited[END_REF]), based on our deterministic scheme, we offer a new perspective on Riemann hypothesis.

II. A NEW AND DETERMINISTIC SCHEME YIELDING THE COMPLETE LIST OF PRIMES UP TO

A PREASSIGNED LIMIT x, WITHOUT ANY PRIMALITY TEST Towards clarifying our approach, let us first make clear our terminology with a few definitions.

Definition 1. N * = N -{0} Definition 2. ∀ p ∈ N, p is a composite odd number if and only if [START_REF] Gauss | Disquisitiones Arithmeticae[END_REF] p is an odd number;

(2) p is not a prime number.

Definition 3. For any n and p elements of N, n and p are consecutive composite odd numbers if and only if there exists no other composite odd number k such that n < k < p.

Our deterministic scheme is built upon the set of composite odd numbers. From these, and as explained below, we deduce non-composite odd numbers i.e prime numbers.

Except 2 and 5, any prime number (PN) has to end with 1 or 3 or 7 or 9. The trivial proof relies on realizing that all other cases are excluded by definition of prime numbers. From the result above we deduce that except 2 and 5, any prime number can be written as Theorem II.1. (i) P N 1 ≡ 10n + 1, ∀n ∈ N, for all primes ending with 1;

(ii) P N 3 ≡ 10n + 3, ∀n ∈ N, for all primes ending with 3;

(iii) P N 7 ≡ 10n + 7, ∀n ∈ N, for all primes ending with 7;

(iv) P N 9 ≡ 10n + 9, ∀n ∈ N, for all primes ending with 9;

Including trivial prime numbers ({2, 3, 5, 7}), the results above states that all prime numbers are also odd numbers. More precisely, all prime numbers are non-composite odd numbers (nocoon in short). For convenience, we will ignore {2, 3, 5, 7} , the trivial prime numbers. We are dominantly interested in the prime numbers greater or equal to 9. Let us therefore introduce the set Ω nocoon of all non-composite odd numbers (i.e prime numbers) greater or equal to 9. Within Ω = {p ∈ N, p ≥ 9 and p is a odd number} , the set that is complementary to Ω nocoon is Ω coon = {p ≥ 9 is a composite odd number}, the set comprising all composite odd numbers greater or equal to 9. Hence Corollary II.2. Ω = Ω coon Ω nocoon with Ω coon Ω nocoon = Ø.

Corollary II.3. All odd numbers are either i Composite or;

ii Prime (non-composite).

From corollary II.2, we read that any odd number that is not composite is therefore a prime number (i.e a noncomposite odd number). By construction, the organization of prime numbers (non-composite odd numbers) is therefore directly correlated to the organization of composite odd numbers. The set Ω coon of composite odd number is at the core of the present work: we deduce all complete list of prime numbers below any preassigned limit s from complete list of composite odd numbers below x.

Lemma 1. Given C, the complete list of all consecutive composite odd numbers below any integer x, any odd number not listed in C is a non-composite odd number i.e. a prime number.

Our deterministic scheme consists therefore in (1) Constructing the complete and ordered list of all composite odd numbers below any preassigned limit x. The tables in corollary II.9 are the backbone of this task;

(2) Based on the test(s) laid down in Theorem II.10 applied to the list obtained in (1) above, one deduces the complete and ordered list of all non-composite odd numbers (i.e prime numbers) smaller than x.

With these preliminaries set, and as a way to illustrate the establishment of our deterministic scheme, let us briefly and momentarily focus on all non-trivial prime numbers (≥ 10) ending with 7 and below any preassigned x, i.e. all prime numbers that can be written as P N 7 and that are smaller than any preassigned limit x. The same reasoning will be applied to all prime numbers that can be written as P N 1 , P N 3 , P N 9 since we established earlier that except 2 and 5, primes have to end with 1 or 3 or 7 or 9. For practical convenience, in all the subsequent sections, 2, 3, 5, 7 will be regarded as trivial prime numbers. The census will therefore be done for all prime numbers larger than 10 and smaller than any preassigned limit x.

We wanted to start by the general problem of listing all the odd numbers ending with 7.

Theorem II.4. Any composite odd number ending with 7 (ii) For any odd number ON 3 ending with 3, ON 3 = 10n 3 + 3, for n 3 ∈ N;

(iii) For any odd number ON 5 ending with 5, ON 5 = 10n 5 + 5, for n 5 ∈ N;

(iv) For any odd number ON 7 ending with 7, ON 7 = 10n 7 + 7, for n 7 ∈ N;

(v) For any odd number ON 9 ending with 9, ON 9 = 10n 9 + 9, for n 9 ∈ N.

Clearly, any odd number ending with 7 can only be written as 10n + 7 where n ∈ N.

Equally trivial is the proof that any composite odd number can only be written as a product of odd numbers (composite or not). If not one ends up with the absurd situation of having the existence of odd numbers also equal to some even number. A straightforward reasoning by the absurd also proves that any finite product of composite odd numbers yields a composite odd number. The latter result is a consequence of the fundamental theorem of arithmetic. Let us therefore list all possible products ON i * ON j for i, j ∈ {1, 3, 5, 7, 9} , of composite odd numbers, since these underlie the decomposition of any composite odd number. There are C 2 5 =

5!

(5-2)!2! such products i.e. fifteen of these products that we list below.

Lemma 3.

P 1 = (10n 1 + 1)(10n 1 + 1) ⇔ P 1 = 10(10n 2 1 + 2n 1 ) + 1, n 1 ∈ N * P 2 = (10n 1 + 1)(10n 2 + 3) ⇔ P 2 = 10(10n 1 n 2 + 3n 1 + n 2 ) + 3, n 2 ∈ N P 3 = (10n 1 + 1)(10n 3 + 5) ⇔ P 3 = 10(10n 1 n 3 + 5n 1 + n 3 ) + 5, n 3 ∈ N P 4 = (10n 1 + 1)(10n 4 + 7) ⇔ P 4 = 10(10n 1 n 4 + 7n 1 + n4) + 7, n 4 ∈ N P 5 = (10n 1 + 1)(10n 5 + 9) ⇔ P 5 = 10(10n 1 n 5 + 9n 1 + n3) + 9, n 5 ∈ N P 6 = (10n 2 + 3)(10n 2 + 3) ⇔ P 6 = 10(10n 2 2 + 6n 2 ) + 9, n 2 ∈ N P 7 = (10n 2 + 3)(10n 3 + 5) ⇔ P 7 = 10(10n 2 n 3 + 5n 2 + 3n 3 + 1) + 5, n 1 ∈ N P 8 = (10n 2 + 3)(10n 4 + 7) ⇔ P 8 = 10(10n 2 n 4 + 7n 2 + 3n 4 + 2) + 1, n 4 ∈ N P 9 = (10n 2 + 3)(10n 5 + 9) ⇔ P 9 = 10(10n 2 n 5 + 9n 2 + 3n 5 + 2) + 7, n 5 ∈ N P 10 = (10n 3 + 5)(10n 3 + 5) ⇔ P 10 = 10(10n The fifteen (15) expressions in Lemma 3 define all the possible forms for any composite odd number. We also know that any composite odd number always ends with 1 or 3 or 5 or 7 or 9. We then deduce from these that all composite odd numbers of the 10n + 7 i.e. ending with 7 are all and only of the type P 4 or P 9 . In other words, Any composite odd number ending with 7 can only be written either as 10n + 7 = (10k 1 + 1) (10k 2 + 7) or as 10n + 7 = (10k 3 + 3) (10k 4 + 9) , where k 2 , k 3 , k 4 are all elements of N and k 1 ∈ N * Similarly, the set of 15 equations in Lemma 3 yields the following results. Equations P 1 and P 15 are special cases of the generic forms for any composite odd number ending with 1. Since we are only interested in the last digit (1), P 1 and P 15 can therefore be extended into P1 and P15 defined by P1 = (10k 1 + 1)(10k 2 + 1) and P15 = (10k 3 + 9)(10k 4 + 9) where k 1 , k 2 , k 3 , and k 4 are all elements of N and k 1 , k 2 ∈ N * . Coupled with P 8 we therefore get that Theorem II.5. All composite odd numbers of the form 10n + 1 i.e. ending with 1 are all and only of the type P1 or P 8 or P15 . In other words, any composite odd number ending with 1 (i.e. of the form 10n + 1) can only be written either as 10n + 1 = (10k 1 + 3) (10k 2 + 7) or as 10n + 1 = (10k 3 + 9) (10k 4 + 9) , or as 10n + 1 = (10k 5 + 1) (10k 6 + 1) , where k 1 , k 2 , k 3 , k 4 are all elements of N and k 5 , k 6 ∈ N * . Generalization of P 1 and P 15 may lead to counting more then once some composite odd number ending with 9. This does not fundamentally alter or affect the fact that we will get final lists made of all consecutive composite odd numbers ending with 1. The same applies to all subsequent situations where duplicates are included in the final lists. We are interested in the final complete list with or without duplicate. Only the fact to it is complete matters.

Theorem II.6. Any composite odd number ending with 3 (i.e. of the form 10n + 3) can only be written either as

10n + 3 = (10k 1 + 3) (10k 2 + 1) or as 10n + 3 = (10k 3 + 7) (10k 4 + 9) , where k 1 , k 3 , k 4 are all elements of N and k 2 ∈ N * .
Theorem II.7. Combining P 3 , P 7 , P 10 , P 11 , and P 15 we obtain the trivial result that any composite odd number ending with 5 (i.e. of the form 10n + 5) can only be written as 10n + 5 = 5 (2k + 1) where k ∈ N.

Theorem II.8. Equations P 6 and P 13 are special cases of the generic forms for any composite odd number ending with 9. Since we are only interested in the last digit (9), P 6 and P 13 can therefore be extended into P6 and P13 defined by P6 = (10k 3 + 3)(10k 4 + 3) and P13 = (10k 5 + 7)(10k 6 + 7) where k 3 , k 4 , k 5 , k 6 are all elements of N. Coupled with P 5 we therefore get that any composite odd number ending with 9 i.e. of the form 10n + 9 can only be written either as 10n + 9 = (10k 1 + 1) (10k 2 + 9) or as 10n + 9 = (10k 3 + 3) (10k 4 + 3) , or as 10n + 9 = (10k 5 + 7) (10k 6 + 7) , where k 2 , k 3 , k 4 , k 5 , and k 6 are all elements of N and k 1 ∈ N * .

The results in theorems II.4, II.5, II.6, II.7 and II.8 define all the possible forms for any composite odd number greater or equal to 10. They can all be summarized in the form of eleven (11) tables given below. Three (03), two (02), one (01), two (02) and three (03) tables are needed for all composite odd numbers ending with 1 or 3 or 5or 7 or 9, respectively. As implicit in corollary II.9, all complete lists of consecutive prime numbers larger or equal to 9 (always ending with 1 or 3 or 7 or 9 ) and below any pre-assigned limit x will be deduced from trivial and systematic operations on elements of these tables, coupled with a test that we establish in Theorem II.10. The reason why some elements above and in the tables are taken in N * is because we are looking for composite odd numbers and the first one of these is 9.

Corollary II.9. Based on the fifteen equations in Lemma 3 and since 5 is the only prime number ending with 5, ten (10 = 11 -1) of the eleven (11) tables given in Appendix A will be effectively used in our deterministic scheme to deduce the complete list of consecutive prime numbers larger than 10 and smaller than any pre-assigned limit x. The ten tables III, IV, V, VI, VII, VIII, IX, X, XI and XII given in Appendix A constitute the numerical backbone of our deterministic scheme. These eleven Tables yield the complete list of all consecutive composite odd numbers greater or equal to 9 and smaller than any pre-assigned limit x. In each table, for each computation p = (10k + m) (10l + q) , k, m, l, q ∈ N, the constrain is that each product p has to be smaller than x.

How then do we deduce, from the previous, the complete list of all consecutive prime numbers below any pre-assigned limit x? The following theorem yields an answer.

Theorem II.10 (Main Theorem). ∀ N 1 and N 2 two consecutive composite odd numbers each larger than 9, ∀ k ∈ N * , Ω k ⊆ N is the interval whose extremities are consecutive and composite odd numbers that are each a multiple of 3. It is trivial to prove that pairs of consecutive odd numbers that are multiple of 3 are separated by a distance of 6. Of all the elements of Ω (x) , those making up Ω k are between pairs of consecutive composite odd numbers that are separated by the smallest distance of all (equal to 6). By construction, any pair of consecutive composite odd numbers within any union of at least two Ω k is separated by a distance strictly larger than 6. As illustrated in Figure 1, intervals of length 6, from any Ω k can be understood as a "unit interval", U: Any interval larger than Ω k has a length that is a multiple of the length of U.

(1) N 2 -N 1 = 2 or N 2 -N 1 = 4 or N 2 -N 1 = 6, (2) If N 2 -N 1 =
(a) We are interested in the length of all intervals between pairs of consecutive composite odd numbers within Ω (x);

(b) Ω k constitutes the shortest interval of integers between pairs of consecutive composite odd numbers within Ω (x) , so it matters;

(c) As illustrated in Figure 1, whatever happens with odd numbers within any Ω k will be mirrored within larger intervals of Ω (x);

(d) As illustrated in Figure 1 and expressed in Lemma 4 below, any number from Ω (x) i.e. any composite odd number will appear within one " unit interval", Ω k .

Let us therefore investigate what happens with numbers within any "unit interval" Ω k .

Lemma 4. ∀p 1 and p 2 ∈ N, two consecutive and composite odd numbers multiple of 3, there always exist two and only two odd numbers between p 1 and p 2 . Each Ω k has a length equal to 6 and can be understood as a "unit interval" of consecutive composite odd numbers. Whatever happens with consecutive composite odd numbers within any Ω k will be mirrored within larger intervals of Ωcoon (x) . Hence the importance of Ω k in quantifying the maximal distance between any pair of consecutive composite odd numbers, since any of these composite odd number will fall within one Ω k .: illustrated here as some of the integers in blue.

We prove it. Let p 1 and p 2 be any two two consecutive and composite odd numbers multiple of 3 : p 11 and p 12 both between p 1 and p 2 are uniquely defined and are both odd numbers. There are three possibilities for p 11 and p 12 : (i) p 11 and p 12 are both composite odd numbers;

p 1 = 3 (2n + 1) , n ∈ N (1)
(ii) one of the two is a prime and the other is a composite number.

(iii) p 11 and p 12 are both prime numbers; Lemma 5. The three options above encompass all possibilities for any pair of composite odd numbers appearing in the ordered set Ω (x) and as illustrated in Figure 1. Let us sum it up. This all happens within every "unit interval" Ω k . Given that Ω (x) = m k=1 k∈N Ω k . Lemma 5 encapsulates the description of the way Ω (x) is systematically populated. Having listed, in Lemma 5, all possible occurrences of pairs of consecutive composite odd numbers within Ω (x) , the constrain emerging is that for any pair (p 1 , p 2 ) of consecutive composite odd numbers, their difference is always smaller or equal to 6. In other words, we have established that ∀ p 1 and p 2 two consecutive composite odd numbers with p 1 < p 2 , one has p 2p 1 ≤ 6.

On the other hand p 1 and p 2 being consecutive, p 2p 1 > 0. This proves the item (1) in Theorem II.10.

(2) Here we prove item (2) in Theorem II.10. We look at all of the only six possibilities. ∀ N 1 ≥ 9 and N 2 ≥ 9 two consecutive composite odd numbers, such that N 2 -N 1 = 2, (i) if N 1 ends with 1 therefore N 2 only ends with 3 and there is no prime number between N 1 and N 2 since the only integer between both will be an even number ending with 2.

(ii) if N 1 ends with 3 therefore N 2 only ends with 5 and there is no prime number between N 1 and N 2 since the only integer between both will be an even number ending with 4.

(iii) if N 1 ends with 5 therefore N 2 only ends with 7 and there is no prime number between N 1 and N 2 since the only integer between both will be an even number ending with 6.

(iv) if N 1 ends with 7 therefore N 2 only ends with 9 and there is no prime number between N 1 and N 2 since the only integer between both will be an even number ending with 8.

(v) if N 1 ends with 9 therefore N 2 only ends with 1 and there is no prime number between N 1 and N 2 since the only integer between both will be an even number ending with 0.

Hence, ∀ N 1 ≥ 9 and N 2 ≥ 9 two consecutive composite odd numbers, If N 2 -N 1 = 2 then there is no prime number between N 1 and N 2 . This also proves that: Lemma 6. ∀ N 1 ≥ 9 and N 2 ≥ 9 two consecutive composite odd numbers, If N 2 -N 1 = 2 then there is one and only one integer between N 1 and N 2 and that integer is even.

(3) Here we prove item (3) in Theorem II.10. Let N 1 ≥ 9 and N 2 ≥ 9 be any two consecutive composite odd numbers such that N 2 -N 1 = 4.

(i)

N 2 = N 1 + 4 (5) = (N 1 + 2) + 2 = Ñ1 + 2 with Ñ1 = N 1 + 2. N 2 = Ñ1 + 2 and Ñ1 = N 1 + 2 yields Ñ1 = N 1 + N 2 2 .
So,

Ñ1 = N 1 + N 2 2 (6) = 2 + N 1 = 2 + (2k 1 + 1) with k 1 ∈ N since N 1 is an odd number = 2 k1 + 1.
In other words, Ñ1 is also an odd integer. Ñ1 and N 2 are therefore two consecutive odd numbers of difference 2. The same goes for N 1 and Ñ1 . From Lemma 6, we deduce that ∃ ! p 1 , p 2 ∈ N all even integers and such that N 1 ≤ p 2 ≤ Ñ1 and Ñ1 ≤ p 1 ≤ N 2 . We also have

N 1 ≤ Ñ1 ≤ N 2 .
In other words there exist three and only three uniquely defined integers p 1 , Ñ1 and p 2 in the interval [N 1 , N 2 ].

Since N 1 ( = Ñ1 ) and N 2 ( = Ñ1 ) are consecutive composite odd numbers and there are only three options for N 2 -N 1 via item (1) in Theorem II.10. We therefore deduce that Ñ1 is a non-composite odd number, otherwise N 1 and N 2 would not be consecutive composite odd numbers as given in Definition 3. We conclude then that Ñ1 is a prime number. This also proves that:

Lemma 7. ∀ N 1 ≥ 9 and N 2 ≥ 9 two consecutive composite odd numbers. If N 2 -N 1 = 4 then there exists one and only one prime number p 1 between N 1 and N 2 . That prime number is given by p 1 = N1+N2 2 .

(4) Here we prove item (4) in Theorem II.10. Let N 1 ≥ 9 and N 2 ≥ 9 be any two consecutive composite odd numbers such that

N 2 -N 1 = 6 ⇔ N 2 -N 1 = 4 + 2 ⇔ N 2 = N 1 + 4 + 2
Here we take advantage of the previous results.

(a)

N 2 = (N 1 + 4) + 2, ( 7
) = Ñ1 + 2 with Ñ1 = N 1 + 4, N 2 = Ñ2 with Ñ2 = Ñ1 + 2.
It is trivial to prove that Ñ1 is an odd integer, as is N 1 . Therefore Lemma 7 states that there exists one and only one prime number p 1 between N 1 and Ñ1 , given by p

1 = N1+ Ñ1 2 ⇔ p 1 = N1+N2 2 -1. (b) Similarly, N 2 -N 1 = 4 + 2 ⇔ N 1 -N 2 = -4 -2.
In other words,

N 1 = N 2 -4 -2 (8) = N2 -2 with N2 = N 2 -4, = N1 with N1 = N2 -2.
It is trivial to prove that N2 is an odd integer, as is N 2 . Therefore Lemma 7 states that there exists one and only one prime number p 2 between N2 and N 2 , given by

p 2 = N2+N2 2 ⇔ p 2 = N1+N2 2 + 1.
This also proves that: Lemma 8. ∀ N 1 ≥ 9 and N 2 ≥ 9 two consecutive composite odd numbers. If N 2 -N 1 = 6 then there exists two and only two prime numbers p 1 and p 2 between N 1 and N 2 . These prime numbers given by p 1 = N1+N2 2 -1 and

p 1 = N1+N2 2 + 1.
Corollary II.9 and Theorem II.10 thus constitute the two foundational components of our deterministic scheme for establishing the complete list of prime numbers up to any preassigned limit x. As evidenced in the results above, no single known primality test is performed. Our scheme operates much like a "compositeness" test. The dramatic gain in computational speed and algorithmic simplicity in obtaining complete lists of small or large prime numbers up to any preassigned limit x opens up unexpected avenues of fertile research on the intrinsic organization of prime numbers and associated. New conceptual developments in the field of number theory are naturally to follow. In Paper II [START_REF] Hawing | Riemann Hypothesis Revisited[END_REF], a follow-up paper to the present one, we illustrate the immense potential attached to our new deterministic scheme by using its key components to offer a new perspective on Riemann Hypothesis.

III. BRIEF NUMERICAL ILLUSTRATIONS AND CONSIDERATIONS

A forthcoming publication will delve further into a few relevant numerical applications of our deterministic scheme. Tables I and II provide a few outputs and statistics for similar runs performed on a high performance computing infrastructure, for values of x ≥ 10 9 . All ten tables in corollary II.9 were used for Table II while only Tables V, VI and VII were used for Table I. The latter highlights how the scheme can be fine-tuned to investigate desired sub-sets of the ensemble of prime numbers, such as all prime numbers ending with either 1, 3, 7, or 9 and all located below any preassigned limit x. This approach may prove useful when, for example, targeting large prime numbers.

By construction, Tables in corollary II.9 yield duplicated values. This does not fundamentally affect the robustness of the final result as these duplicates are easily removed. The final lists typically encompass all the possibilities. The final list is then ordered. We are left with adding -by hand -the very few trivial ones, which by construction of the tables, are not listed since the smallest values from each one of the table is set by the product of the two integer appearing in each factor in the the product (10k + l) (10m + n) , i.e. the product mn. Shown in Table I are a few statistics from preliminary runs on facilities hosted by the South African Centre for High Performance Computing2 in Cape Town, South Africa. The results shown are for composite odd numbers ending with 1. We now know that within our approach, sets made of consecutive composite odd numbers are the gateways into prime numbers. Not fully disclosed here is that computational options exists to restrain the search for both composite and therefore primes within any preassigned interval [x 1 , x 2 ] with x 1 , x 2 ∈ N. This will figure in a forthcoming paper. When compared with Table II, the preliminary execution times given in Table I suggests forthcoming breakthroughs towards computing -much faster and much simpler -prime numbers, small and large, via a deterministic way. Table II gives a few statistics from preliminary runs on the South African Centre for High Performance Computing infrastructure. In Figures 2 and4, we show evolutions of the values of π(x) and ρ(x) = π(x) x for bin [10 5 k, 10 5 (k + 1)], with k = 1, ..., 1000. On Figures 3 and5, one notes the same trend as the well-known function ρ(x) ≡ 1 log(x) , approximating ρ(x) and that can be deduced from the prime number theorem [START_REF] Hadamard | [END_REF][START_REF] De La La Vallée-Poussin | Article in a journal[END_REF]. A forthcoming publication will delve further into a few relevant numerical applications of our deterministic scheme. The reader is advised to write us at the email provided to gain access to some of the complete list mentioned in the tables below. Complete lists of prime numbers up to 10 9 are freely available on this website http://univgandhi-guinee.com/algo/public/home/nombres-premiers3 . We have established a deterministic scheme that yields the complete list of prime numbers below any preassigned limit x. We have observed a considerable gain in computational speed and algorithmic simplicity towards obtaining complete lists of prime numbers both small or large. Our deterministic scheme shines a new and unique light on the importance of the set of composite odd numbers. From these composite odd numbers, non-composite odd number i.e prime numbers are deduced. At the core of the deterministic scheme is a set of eleven (11) generic tables fully characterized and leading to complete lists of ordered composite odd numbers below x. The said tables are coupled with a three-criteria systematic test applied on differences between pairs of the consecutive composite odd numbers obtained. Counting all the elements of the obtained list of prime numbers therefore provides a new and fertile approach in tackling the long standing problem of "how many prime numbers are there below any preassigned limit x. Our Evident here is the well known increased scarcity of prime numbers as the preassigned limit x increases.

scheme presented here suggests a solution to that problem in a simple, coherent and deterministic way. In a follow-up paper [START_REF] Hawing | Riemann Hypothesis Revisited[END_REF], we propose -based on our scheme -a new perspective on Riemann hypothesis. Thus highlighting the immense potential, for number theory and related fields, embedded in the new approach presented in this work.

Definition 4 .

 4 Ω nocoon = {p = 10n + k with n ∈ N * , p non-composite odd number, and k ∈ {1, 3, 7, 9}}.

Lemma 2 .

 2 ∀ p ∈ Ω coon , ∃ s 1 , s 2 , ...., s n each in Ω coon such that p = s 1 * s 2 * ... * s n . By consequence, one can always write p = k * l * m where l and m are each one a composite odd number and k ∈ N

2 3 +

 3 10n 3 + 2) + 5, n 3 ∈ N P 11 = (10n 3 + 5)(10n 4 + 7) ⇔ P 11 = 10(10n 3 n 4 + 7n 3 + 5n 3 + 3) + 5, n 4 ∈ N P 12 = (10n 3 + 5)(10n 5 + 9) ⇔ P 12 = 10(10n 3 n 5 + 9n 3 + 5n 5 + 4) + 5, n 5 ∈ N P 13 = (10n 4 + 7)(10n 4 + 7) ⇔ P 13 = 10(10n 2 4 + 14n 4 + 4) + 9, n 4 ∈ N P 14 = (10n 4 + 7)(10n 5 + 9) ⇔ P 14 = 10(10n 4 n 5 + 9n 4 + 7n 5 + 6) + 3, n 5 ∈ N P 15 = (10n 5 + 9)(10n 5 + 9) ⇔ P 15 = 10(10n 2 5 + 18n 5 + 80) + 1, n 5 ∈ N

  2 then there is no prime number between N 1 and N 2 , (3) If N 2 -N 1 = 4 then there is one prime numbers between N 1 and N 2 , (4) If N 2 -N 1 = 6 then there are two prime numbers between N 1 and N 2 . Proof. (1) We prove item (1) in Theorem II.10. The result is immediate for any composite odd number from 1, 3, 5, 7. Let us introduce the ordered set Ω (x) of all composite odd numbers smaller or equal to than x ∈ N. By construction, 9 is the first element of Ω (x) . Ω (x) = m k=1, k∈N Ω k and Ω k = {p ∈ N, p is a composite odd number and 3 (2k + 1) ≤ p ≤ 3 (2k + 3) , k ∈ N * } ⊆ N Ω (x) can be visualized as the line of integers (consecutive composite odd numbers) shown in Figure 1.

FIG. 1 :

 1 FIG.1: Partially illustrated here is the ordered set of all composite odd numbers multiple of 3 and smaller or equal to some x ∈ N (upper ticks in black). By construction, 9 is the first (smallest) composite odd number. Ωcoon (x) is the set of all consecutive composite odd numbers smaller or equal to x. Ωcoon (x) =

  (a) If p 11 = 6n + 5 is a composite odd number (and p 12 = 6n + 7 is a composite odd number) then p 1 = 6n + 3 and p 11 are consecutive composite odd numbers and their difference is equal to 2; (b) If p 12 = 6n + 7 is a composite odd number and p 11 = 6n + 5 is a prime number then p 1 = 6n + 3 and p 12 = 6n + 7 are consecutive composite odd numbers and their difference equal to 4; (c) p 11 = 6n + 5 and p 12 = 6n + 7 are prime numbers with p 1 = 6n + 3 and p 2 = 6n + 9 consecutive and composite odd number. The difference p 2p 1 = 6.

FIG. 2 :

 2 FIG.2: Shown here is the evolution of the cumulative number of primes over sets of bin [(k -1) * 10 5 , k * 10 5 ] for k ∈ N, 2 ≤ k ≤ 1000.

FIG. 3 :FIG. 4 :

 34 FIG.3: Shown here is the density from the cumulative number of primes over sets of bin [(k -1) * 10 5 , k * 10 5 ] for k ∈ N, 2 ≤ k ≤ 1000. Visible here and in Figure5is the well-known function ρ(x) ∝ 1 log(x) , approximating ρ(x) that can be deduced from the prime number theorem[START_REF] Hadamard | [END_REF][START_REF] De La La Vallée-Poussin | Article in a journal[END_REF]

  FIG. 5: Shown here is the evolution of the number of primes per bin number [(k -1) * 10 5 , k * 10 5 ] for k ∈ N, 2 ≤ k ≤ 1000.

  , n 3 , n 4 are all elements of N and n 2 ∈ N * . For any odd number ON 1 ending with 1, ON 1 = 10n 1 + 1, for n 1 ∈ N;

	1. is of the form 10n + 7, for n ∈ N;
	2. can only be written either as
	(i) 10n + 7 = (10n 1 + 7) (10n 2 + 1);
	(ii) or as or as 10n + 7 = (10n 3 + 3) (10n 4 + 9) ,
	where n, n 1 Proof. It is trivial to prove that any odd number (ON) ends with 1 or 3 or 5 or 7 or 9, since the excluded cases
	(0 or 2 or 4 or 6 or 8) lead to even numbers. Hence
	(i)

TABLE I :

 I A few statistics from preliminary runs on a high performance computing infrastructure. Shown below are results for composite odd numbers ending with 1. When compared with TableII, the preliminary execution times given in this table suggests potential breakthroughs towards computing -much faster and mush simpler -prime numbers, small and large.

	Composite odd number ending with 1 and below x = Execution time on a high performance computing infrastructure
	10 9	01 hour 10 minutes
	5 * 10 9	43 minutes
	10 * 10 9	02 hours

TABLE II :

 II A few statistics from preliminary runs on a high performance computing infrastructure: Shown below are results for prime numbers below a preassigned limit x. These are all deduced from associated list of composite odd numbers below x.

	Prime numbers below x = Number of prime numbers below x Execution time on a high performance computing infrastructure
	0.1 * 10 9	5761455	5 minutes
	0.25 * 10 9	13679318	14 minutes
	0.5 * 10 9	26355867	31 minutes
	10 9	50847534	01 hour 53 minutes
		IV. CONCLUSION AND FUTURE WORK

TABLE III :

 III Simplifying representation in table for {(10k1 + 1) (10k2 + 7) , k2 ∈ N and k1 ∈ N * } leading to composite odd numbers ending with 7

	k1 k2 (10k1 + 1) (10k2 + 7)
	1 0	11	7
	2 1	21	17
	.. ...	...	...
	∞ ∞	∞	∞

TABLE IV :

 IV Simplifying representation in table for {(10k3 + 3) (10k4 + 9) , k3, k4 ∈ N} leading to composite odd numbers ending with 7

	k3, k4 (10k3 + 3) (10k4 + 9)
	0	3	9
	1	13	19
	2	23	29
	...	...	...
	∞	∞	∞

TABLE V :

 V Simplifying representation in table for {(10k1 + 3) (10k2 + 7) , k1, k2 ∈ N} leading to composite odd numbers ending with 1

	k1, k2 (10k1 + 3) (10k2 + 7)
	0	3	7
	1	13	17
	2	23	27
	...	...	...
	∞	∞	∞

TABLE VI :

 VI Simplifying representation in table for {(10k3 + 9) (10k4 + 9) , k3, k4 ∈ N} leading to composite odd numbers ending with 1

	k3, k4 (10k3 + 9) (10k4 + 9)
	0	9	9
	1	19	19
	2	29	29
	...	...	...
	∞	∞	∞

TABLE VII :

 VII Simplifying representation in table for {(10k5 + 1) (10k6 + 1) , k5, k6 ∈ N * } leading to composite odd numbers ending with 1

	k5, k6 (10k5 + 1) (10k6 + 1)
	1	11	11
	2	21	21
	...	...	...
	∞	∞	∞

TABLE VIII :

 VIII Simplifying representation in table for {(10k1 + 3) (10k2 + 1) , k1, k2 ∈ N and k2 ∈ N * } leading to composite odd numbers ending with 3

	k1 k2 (10k1 + 3) (10k2 + 1)
	0 1	3	11
	1 2	13	21
	.. . ...	...	...
	∞ ∞	∞	∞

TABLE IX :

 IX Simplifying representation in table for {(10k3 + 7) (10k4 + 9) , k3, k4 ∈ N} leading to composite odd numbers ending with 3

	k3, k4 (10k3 + 7) (10k4 + 9)
	0	7	9
	1	17	19
	...	...	...
	∞	∞	∞

TABLE X :

 X Simplifying representation in table for {(10k1 + 1) (10k2 + 9) , k1, k2 ∈ N and k1 ∈ N * } leading to composite odd numbers ending with 9

	k1 k2 (10k1 + 1) (10k2 + 9)
	1 0	11	9
	2 1	21	19
	... ...	...	...
	∞ ∞	∞	∞
	TABLE XI: Simplifying representation in table for {(10k3 + 3) (10k4 + 3) , k3, k4 ∈ N} leading to composite odd numbers
	ending with 9
	k3, k4 (10k3 + 3) (10k4 + 3)
	0	3	3
	1	13	13
	...	...	...
	∞	∞	∞

TABLE XII :

 XII Simplifying representation in table for {(10k5 + 7) (10k6 + 7) , k5, k6 ∈ N} leading to composite odd numbers ending with 9

	k5, k6 (10k5 + 7) (10k6 + 7)
	0	7	7
	1	17	17
	...	...	...
	∞	∞	∞

TABLE XIII :

 XIII Simplifying representation in table for {5 (2k + 1) , k ∈ N} leading to composite odd numbers ending with 5

	k 5 (2k + 1)
	0	5
	1	15
	2	25
	...	...
	∞	∞

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Acknowledgements:

G.H acknowledges support from "L'Université Privée Mahatma Gandhi" in Conakry and the " Ministère de la Recherche Scientifique " of the Republic of Guinea. PO is funded by the South African National Research Foundation (NRF). PO acknowledges the support and the generous hospitality provided by the "L'Université Mahatma Gandhi" in Conakry and the " Ministère de la Recherche Scientifique " of the Republic of Guinea during the research visits towards the present work. The authors wish to express their immense gratitude to Fode Idrissa Soumare for the critical support provided on the computational component of the present work. Our words of thanks are also addressed to Lansana Bangoura for his comments. We acknowledge use of the facilities of the Centre for High Performance Computing in Cape Town, South Africa.

Appendix A: The Eleven Generic Tables for Our Deterministic Scheme

In each table and for each computation of p = (10k + m) (10l + q) , k, m, l, q ∈ N, the constrain is that each product p has to be smaller than x, the preassigned limit.