Imen Trabelsi
email: imen.trabelsi@supmeca.fr

Marc Zolghadri
email: marc.zolghadri@supmeca.fr

Besma Zeddini

Maher Barkal- Lah

Mohamed Haddar

FMECA-based risk assessment approach for proactive obsolescence management

Keywords: Product Lifecycle Management (PLM), Proactive Obsolescence Management, Risk Assessment, Weibull

One of the most critical issues that can affect the useful life of a product is the obsolescence of its components or functionalities. To minimize its effects and bring long-term benefits to systems, obsolescence must be proactively managed. A critical step in the process of proactive obsolescence management is the obsolescence risk analysis of critical components or functionalities. However, estimating the degree of obsolescence of multi-component systems is an area that is still under-explored, particularly when interdependencies exist between components. This estimation can be more complicated where there is no prior knowledge about the interaction between components. We used Weibull's distribution to model the components' interaction and calculate the obsolescence degree of the global system. This approach is evaluated using a numerical example based on a meteorological data acquisition system. The obsolescence of main components of this system is modeled while taking into account the interaction between them. Results are presented and discussed while clarifying the scientific issues that should be tackled in future works.

Introduction

Nowadays, obsolescence has become an increasingly common problem in all sectors, whether in aeronautics, automotive, home automation, etc. This problem is mainly related to systems with a long-life cycle [START_REF] Zolghadri | Obsolescence prediction: a Bayesian model[END_REF]. The system is designed to operate for a long period while some of its components are exposed to some kind of obsolescence and therefore the vulnerability and exposure to obsolescence of the system increases. Obsolescence or Diminishing Manufacturing Sources and Material Shortages (DMSMS) is defined as "the loss or imminent loss of original manufacturers of items or suppliers of items or raw materials" [START_REF] Sandborn | Designing for technology obsolescence management[END_REF]. In [START_REF]Diminishing Manufacturing Sources and Material Shortages (DMSMS) A Guidebook of Best Practices for Implementing a Robust DMSMS[END_REF], the authors clarified the difference between these two concepts by explaining that "obsolescence does not always lead to DMSMS problems, and not all DMSMS problems are due to obsolescence". As long as the product is still in production or available to meet future demands, it does not pose a DMSMS problem. But, if the spare parts are obsolete or if the know-how to perform the repair and the possibility to test the system (e.g. test equipment) after repair are not available, this causes a DMSMS problems. The authors in [START_REF] Pecht | Electronic part life cycle[END_REF] explained the difference between obsolescence and discontinuance. By specifying that obsolescence occurs at a technological level, whereas discontinuance occurs when the manufacturer stops producing an item. However, in the industrial domain, obsolescence includes the cessation of production of a component if there are no other manufacturers for that specific component. This paper deals with the obsolescence problem which can be due to the technological evolution or the absence of support from manufacturers. The objective is to minimize its impact on life cycle costs, system performance, availability, maintainability and safety. Thus, it is necessary to focus on obsolescence management throughout the product life cycle. In [START_REF]Diminishing Manufacturing Sources and Material Shortages (DMSMS) A Guidebook of Best Practices for Implementing a Robust DMSMS[END_REF], the authors present a multidisciplinary process for obsolescence management aimed at identifying the problems resulting from obsolescence; assessing their impacts; analyzing mitigation actions, and finally implementing the most costeffective strategy.

Obsolescence syndrome concerns all kinds of systems, but its impact is more important in complex systems [START_REF] Meyer | A model to manage electronic component obsolescence for complex or long life systems[END_REF]. Indeed, complex systems are more exposed to obsolescence problems due to their multi-connected components. A complex system can be defined as a coherent set of different interconnected entities [START_REF] Bian | Stochastic modeling and real-time prognostics for multicomponent systems with degradation rate interactions[END_REF]. Thus, the study of its obsolescence depends on the obsolescence of each entity and the interactions between them. Components dependency refers to the fact that the obsolescence of one component can influence the life span of the other components of the system. In other words, if a component has become obsolete for various reasons, it may have an impact on the other components, which interact with it, by accelerating their obsolescence. Only a few studies, in the obsolescence literature, have been considered the complexity of systems and the interaction between their components [START_REF] Soltan | System Engineering for dependency analysis-a Bayesian approach: application to obsolescence study[END_REF][START_REF] Carmignani | An integrated structural framework to cost-basedfmeca: The priority-cost fmeca[END_REF]. This problem becomes more difficult when considering the obsolescence evolution over time. Up to our knowledge, conducted works and existing methods in the obsolescence domain do not address the evolution of obsolescence over time. Our research aims to develop an approach to predict system obsolescence over time while taking into account the interactions between its components. To do, the Weibull distribution is used to model the obsolescence evolution of components. Based on these distributions, the system obsolescence degree is calculated while taking into account the interactions of its components.

The remainder of this paper is organized as follows: Section 2 presents a brief overview of obsolescence concepts and reviews the different methods of obsolescence management. Section 3 describes the proposed methodology with the necessary assumptions. Under these assumptions, we illustrate our approach in section 4 by an example. Finally, conclusions and perspectives are given in section 5.

Obsolescence State of the Art

In this document and in its context, we adopt the definition of obsolescence given as "the loss or imminent loss of system functions due to the loss of items original manufacturers, items or raw materials suppliers, or technical support". The problem of obsolescence is particularly well illustrated by electronic components, but it is not limited to them. Authors in [START_REF] Rojo | Obsolescence management for long-life contracts: state of the art and future trends[END_REF] gave a global view of obsolescence, specifying that obsolescence can affect other aspects such as mechanical components, software, materials, skills, tools, and test equipment. Obsolescence goes through different stages; upon notification of change or discontinuity, the company must immediately inform its customers so that they can prepare their obsolescence management plan [5]. Obsolescence can be relative or absolute [START_REF] Granberg | The quality re-evaluation process: product obsolescence in a consumerproducer interaction framework[END_REF]. When the product stops working (a technical problem), obsolescence is said to be absolute. If it still works and consumers, for example, have chosen to replace it, obsolescence is said to be relative [START_REF] Cooper | Inadequate life ? Evidence of consumer attitudes to product obsolescence[END_REF].

Obsolescence Causes

There are many possible reasons for obsolescence. Some of them are related to technological evolution or innovation that makes a component obsolete even though it still works and can be produced and purchased. Other reasons are related to the supplier, such as the disappearance of original suppliers from the market for various reasons [START_REF] Atterbury | Component Obsolescence Group, United Kingdom; Minenfeld Obsolescence, A Guide to Tackling Disappearing Products[END_REF], or the original component manufacturer or original equipment manufacturer is not willing to continue producing a part for economic reasons for example; when the demand for a component or technology decreases and the manufacturer considers it uneconomic to continue production. Obsolescence can be caused by directives, rules and other laws imposed by governments such as the restriction of the use of certain hazardous substances (RoHS) directive [START_REF] Brewin | The Pb-free minefield: a guide to mitigating risk during the transition to RoHS compliance[END_REF] such as the ban on the use of Freon because of its ozonedepleting characteristics. Products may also become obsolete if they are no longer durable due to the ageing process of stored parts, which make their use impossible [START_REF] Bartels | Strategies to the prediction, mitigation and management of product obsolescence[END_REF].

Obsolescence Consequences

Obsolescence is a significant cost generator and can affect products during all their life cycles. Obsolescence consequences can be divided into three main categories: (i) product's requirements, (ii) ecologic and (iii) economic consequences. Requirements consequences imply the no respect of system requirements which can be classified according to [START_REF] Glinz | On non-functional requirements[END_REF], in four categories, namely; functional system requirements (the suitability of the component to fulfill its function in its environment); non-functional requirements (maintainability, viability, vulnerability, testability...); performance requirements (efficiency, execution time, transport speed...); constraints (dimensions, supply voltages...). Moreover, obsolescence has strong effects on the environment. In fact, obsolete products result in a huge volume of hazardous waste that intensify the greenhouse gas emissions and pollution. In addition, the replacement of these obsolete products implies an increased use of non-renewable resources. From economical point of view, obsolescence can result in an important life cycle costs that call into question the depreciation of systems. There is a clear evidence that a robust obsolescence management strategy can significantly reduce the impact of these consequences. Thus, many obsolescence management strategies have been proposed in the literature.

Obsolescence Management

While obsolescence is inevitable [5], its anticipation with a careful planning can minimize its impact and potentially high cost. Sandborn [START_REF] Sandborn | Design for obsolescence risk management[END_REF] has defined three types for obsolescence management: (i) Reactive management which involves taking action when obsolescence has already occurred, (ii) proactive management is put in place for critical components that are at risk of be-coming obsolete and (iii) strategic management complements proactive and reactive management and includes determining the optimal combination of mitigation approaches and design refreshes.

To carry out these management strategies resolving and mitigating methods are used. Resolving techniques are applied to solve the obsolescence problem when it arises. While mitigation methods are applied proactively to minimize the impact of obsolescence. Table 1 identifies these methods. Table 1. Methods for resolving and mitigating obsolescence [START_REF] Rojo | Obsolescence management for long-life contracts: state of the art and future trends[END_REF].

Method Description

RESOLUTION

Redesign

Redesign obsolete parts to improve the performance, reliability, and maintainability of the system.

Last Time Buy

Purchase and store components following a supplier's product discontinuation notice.

FFF equivalent

Replace the obsolete component with a component that will work fully (in terms of form, fit, and function)

Emulation Develop an item with identical form, fit, and function to replace the obsolete parts using advanced technologies.

Cannibalization

Keep products by using components and subassemblies from other unserviceable systems.

MITIGATION Lifetime buy

Purchase and store components and spare parts, before production, from the original component manufacturer to meet the system's needs.

Use Multisourced Components

Check that the components included in the BOM can be provided by several suppliers to minimize the number of critical components. Applicable at the design stage Technology Roadmapping Help to identify, evaluate and select technological alternatives in order to solve and prevent obsolescence problems caused by technological change.

The most commonly used type of management is the reactive one because of its easy implementation and also because companies are often far from being able to implement any proactive or strategic managements. However, it is advisable to use it only if the cost associated to the obsolete component is low [START_REF] Pingle | Selection of obsolescence resolution strategy based on a multi criteria decision model[END_REF]. However, if the obsolescence likelihood and the associated costs are high, proactive obsolescence management strategies are recommended to minimize the risk of obsolescence and the associated costs [START_REF] Meng | Strategic proactive obsolescence management model[END_REF][START_REF] Rojo | Obsolescence management for long-life contracts: state of the art and future trends[END_REF]. Various authors [START_REF] Meyer | A model to manage electronic component obsolescence for complex or long life systems[END_REF][START_REF] Sandborn | A Data Mining Based Approach to Electronic Part Obsolescence Forecasting[END_REF] have advised the use of obsolescence surveillance to obtain early notification of any risk of obsolescence. For that purpose, in this paper we focus on the proactive obsolescence management by proposing a risk analysis tool to predict the obsolescence over the time.

Methodology and Tools

Dealing with obsolescence involves a lot of trade-offs, risk assessment, and long-term planning. The key enabler to successful proactive obsolescence management is to predict the exposure degree to obsolescence. The objective of this section is to describe our proposed approach to predicting the evolution of obsolescence over time by taking into account the interactions between components. As shown in Fig. 1, the proposed method consists of three main steps:

1. Obsolescence risk analysis: in this step, the FMECA tool is used to identify main components and their exposure to obsolescence. The FMECA tool is used because it is extremely widespread in all domains that affect the safety, reliability, availability, and maintainability of a product or system [START_REF] Carmignani | An integrated structural framework to cost-basedfmeca: The priority-cost fmeca[END_REF]. 2. Obsolescence modelling: The Weibull distribution is chosen to model the evolution of obsolescence over time because of its generality and flexibility. 3. Manage component interactions.

FMECA-Obso: Risk Analysis

Obsolescence risk assessment using the Failure Modes, Effects and Criticality Analysis (FMECA-Obso) tool is conducted to analyze the sensitivity of the product to obsolescence and to prioritize risks to manage the components proactively or reactively. Thus, critical entities (components, functions and functionalities) with a high risk of obsolescence are identified, prioritized and managed throughout the system lifecycle. The FMECA-Obso criteria are presented below: ─ Gravity: Gravity in this sense refers to the level of risk of the effect of obsolescence.

The point here is to prioritize the gravity of the effects of obsolescence. ─ Occurrence: The index of occurrence refers to the degree of occurrence of product obsolescence, which depends primarily on the characteristics and functions of the product. For microelectronic components, the occurrence can be classified as frequent because these components are highly subject to technological development. ─ The detection: The assigned index is used to identify the non-detectability of obsolescence.

The evaluation of the obsolescence risks for each entity results in the calculation of the Degree of Obsolescence DO, based on the estimation of the Gravity, Occurrence and Detection indices. An expert on the system domain fixes these parameters.

Prediction Method

In this part, we are interested in predicting the obsolescence of systems. Predicting is seen as a scientific challenge for the implementation of such a proactive management strategy in which estimating the date of obsolescence is important. To do, we consider the degree of obsolescence (DO) as a Weibull distribution. This probabilistic model is used due to its generality, flexibility and wide applicability in the system life cycle domain [START_REF] Weibull | Wide applicability[END_REF]. The Weibull probability density function is given by equation (1):

f(t, θ, k, λ) = k λ ( t-θ λ ) k-1 ⅇ -( t-θ λ ) k (1)
where t, θ, k, and λ are strictly positive parameters. k and λ are the shape and scale parameters respectively, and θ presents the position parameter. These parameters define the probability distribution and model component obsolescence. We assume that the predictions were done at the same time for all system components, i.e. θ = 0.

To estimate the Weibull parameters from the FMECA indices, we assume that the shape parameter k is proportional to the indices of gravity and non-detection. In fact, more gravity is high and there is no way to detect the problem (high non-detection index), more the risk of exposure to obsolescence is high: 𝑘 = 𝑎. 𝐺. 𝐷 where a is a constant, G and D are the indices of gravity and detection.

As for the scale parameter λ, it is inversely proportional to the occurrence index; because with more frequent occurrence, the risk of obsolescence is accelerated (λ small): 𝜆 = 𝑏 𝑂 where b is a constant and O is the index of occurrence.

The degree of obsolescence DO is therefore given by the following formula: To analyze the interactions between the components and to correctly monitor their state of obsolescence, it is good practice to represent each component by its cumulative distribution. The cumulative distribution function CDF of Weibull's law is:

𝐹(𝑡; 𝑘; 𝜆) = 1 -ⅇ -( 𝑡 𝜆 ) 𝑘 (3)
Interaction Rate. The interaction principle considers that the obsolescence of one component induces the obsolescence of the other components that interact with it. The state of obsolescence of the affected component Ca is accelerated with an interaction rate 𝜌 𝑖→𝑎 , which generally depends on the state of the influencing component Ci.

The interaction rate is given by the following formula:

𝜌 𝑖→𝑎 = 1 + 𝐹 𝑖 (𝑡,𝜆 𝑖 ,𝑘 𝑖 ) 1-𝐹 𝑎 (𝑡,𝜆 𝑎 ,𝑘 𝑎 ) (4)

Fig. 2. CDF with different obsolescence states

Consider s(t) the function that indicates the obsolescence status of the component, e.g. "nominal", "critical" or "obsolete" (see Fig. 2) given by:

𝑠(𝐹(𝑡)) = { 1 𝐹(𝑡) < 𝑝 1 2 𝑝 1 < 𝐹(𝑡) < 𝑝 2 3 𝐹(𝑡) > 𝑝 2 (5)
Weibull Model Taking into Account Interactions. In characterizing the effects of the interaction between entities in the system, the dependence model is described by the evolution of the obsolescence of the CDF as follows:

𝐹(𝑡; 𝑘; 𝜆) = 1 -ⅇ -( 𝑡 𝜆 ′ ) 𝑘 (6)
where λ' =

𝜆 𝜌(𝑡)

To set up this model, we consider the following assumptions:

─ Without taking into account the dependency, the Weibull parameters are calculated from the Gravity, Occurrence and Detection indices by the following formulas:

𝑘 = 𝐺𝑟𝑎𝑣𝑖𝑡𝑦 * 𝐷ⅇ𝑡ⅇ𝑐𝑡𝑖𝑜𝑛 ; 𝜆 = 1 𝑂𝑐𝑐𝑢𝑟𝑟𝑒𝑛𝑐𝑒 (7)
─ The analysis of the degrees of obsolescence for all system components was done at the same time. ─ The rate of interaction depends on the state of obsolescence of the influencing component. ─ Obsolescence is assumed to be unidirectional (No mutual interaction).

4

Case Study

Description of the Case

The example illustration used is an environment observation link to earth (EOLE) system (Fig. 3), a weather balloon system whose main purpose is to provide meteorological data to various scientific users. This system is used as a case study in various previous works on obsolescence, as in [START_REF] Soltan | System Engineering for dependency analysis-a Bayesian approach: application to obsolescence study[END_REF]. It is composed of two subsystems; the first represents the in-flight acquisition subsystem, it is composed of a Helium-filled balloon, a temperature sensor, a pressure sensor, a sensor carrier (nacelle) and a nanocomputer where the data provided by the sensors are collected. This computer will, in turn, transmit them to the processing subsystem on the ground.

In our first work, we dealt with the obsolescence of VHF transmission technology. It was pointed out that the upgrade from VHF to UHF will result in changes to the physical architecture of the system, in particular the replacement of the radio transmitter and receiver. We assume that the transmission technology and the transceiver are the most critical components of the system where the antenna is the influencing component and the transceiver is the affected component. The obsolescence interaction is manifested in the cumulative distribution functions of these interdependent components.

To show the impact of the interaction on the obsolescence process of the system, we will consider two simulations. In the first simulation, the model does not take into account the interactions between the components. While in the second simulation, the interactions between the components of obsolescence will be studied. After a first analysis FMECA_Obso, the estimates of the Gravity, Occurrence and Detection indices are listed in Table 2.

The components are presented by the probability density functions fVHF(t) and fRadio(t) quoted in [START_REF] Atterbury | Component Obsolescence Group, United Kingdom; Minenfeld Obsolescence, A Guide to Tackling Disappearing Products[END_REF], where k and λ are the Weibull parameters and to observation time. Fig. 3. Simplified specifications of the "EOLE" system [START_REF] Roques | Systems Architecture Modeling with the Arcadia Method: A Practical Guide to Capella[END_REF] According to Weibull's model, obsolescence is well modeled by the cumulative function F(t) as in Fig. 4. It is assumed that initially the interaction rate is equal to 1: 𝜌 𝑖→𝑎 = 1 which implies that the interactions between the components have not been taken into account. Generally, the rate of interaction depends on the state of obsolescence of the influencing component. It is very important when the state of obsolescence of VHF technology is serious. To study the impact of the interaction rate on the degree of obsolescence, we consider two levels for the interaction rate; 𝜌 𝑖→𝑎 = 2 when the obsolescence status of the VHF is critical and 𝜌 𝑖→𝑎 = 3 when the VHF is considered obsolete (see Fig. 5).

Results and Discussions:

The degree of obsolescence of the system increases as the rate of interactions increases and this is well illustrated in Table 3. Regardless of the inter-dependency, VHF technology is considered the most critical element that its obsolescence causes system obsolescence. However, considering the interactions between the components, the degree of obsolescence of the system will depend on the degree of obsolescence of the radio transceiver. Fig. 6 shows the change in the degree of radio obsolescence. In this example, the Weibull distribution quoted in [START_REF] Granberg | The quality re-evaluation process: product obsolescence in a consumerproducer interaction framework[END_REF] was considered to vary the scale parameter as a function of the interaction rate. But really, the shape parameter also influences the variation in the obsolescence degree. Hence, in future work, we will consider that the shape parameter k will be affected by the interaction between the components.

Fig. 6. Obsolescence degree of radio taking into account interaction Conclusions and Outlook

In this paper, we presented FMECA-Obso for analyzing the risks of obsolescence. This tool consists of an in-depth analysis of the system to identify critical components and functions that can lead to system obsolescence. We also presented a model for predicting the obsolescence of a complex system where there are interactions between its components. From FMECA-Obso we can calculate both the degree of obsolescence and focus on the effect of the interaction between components on system obsolescence. The model is illustrated with a numerical example with a comparison between the different degradation models.

In this study, we considered that obsolescence is unidirectional, but in practice the interaction between the components could be mutual, i.e. obsolescence can propagate in both directions. Thus, not all components present the same risk of obsolescence and some are resilience, but they remain vulnerable. The challenge is to find a model for each component that integrates all these factors. These are our goals in future work to better align this obsolescence prediction using machine-learning techniques that could increase the prediction accuracy.

Fig. 1 .

 1 Fig. 1. Illustration of the FMECA-Obso methodology

  the date of observation.

Fig. 4 .

 4 Fig. 4. CDF of the components Fig. 5. Obsolescence states of VHF

Table 2 .

 2 Obsolescence risk analysis FMECA_OBSO

	Temporal Horizon: 2 years		Obso. value
	Entities	Availability or Suitability	Obsolescence classification	Effect	G	O	D
	Technology VHF	Suitability	Technological	Syst. usability	9	7	1
	Radio Emitter and Receiver	Suitability	Functional	Syst. performance	6	7	1

Table 3 .

 3 Simulation results

	𝜌 𝑖→𝑎	DO VHF	DO Radio	DO System
	1	0.8	0.75	0.8
	1.5	0.8	0.98	0.98
	2	0.8	1	1


Frédéric Jiguet
email: frederic.jiguet@mnhn.fr

Kevin J Kardynal

Keith A Hobson

Feather stable isotope (δ 2 H) measurements suggest no historical variation in latitudinal origin of migrants in two declining songbirds

Keywords: Brambling, Emberiza hortulana, climate change, deuterium, Fringilla montifringilla, northward shift, Ortolan Bunting

The geographic origin of migratory birds can be traced using measurements of stable-hydrogen isotopes (δ 2 H) in their feathers. In continental Europe, local amount-weighted mean annual δ 2 H in precipitation varies with latitude, so that assignments of geographical origin of bird feathers are linked to latitude. Consequently, a potential northward latitudinal shift in the distribution of a species' abundance is expected to translate into a shift in δ 2 H values for feathers grown on the breeding grounds. We analyzed δ 2 H in feathers from migrant individuals of Brambling (Fringilla montifringilla) and Ortolan Bunting (Emberiza hortulana) collected in France from the mid-19 th century to recent years (2015), to assess potential temporal trends in feather δ 2 H that may indicate changes in source populations or a shifting breeding range. These two species have shown long-term population declines in Europe with subsequent range retractions, but we did not find obvious differences in feather δ 2 H values in samples collected over the last 160 years. One potential explanation for this result is that population declines occurred similarly at all latitudes across the breeding ranges without involving a measurable latitudinal shift in abundance that could translate into changes in feather δ 2 H values.

However, this result may also be due to potential temporally shifting terrestrial spatial δ 2 H gradients (i.e. isoscape) similar to a shift in bird abundances, or to limited northward range shifts that do not translate into detectable trends in feather δ 2 H.

Introduction

Many common breeding bird species are declining across the northern hemisphere because of global change, including climate change (Stephens et al. 2016[START_REF] Rosenberg | Decline of the North American avifauna[END_REF]. Climatic change has been linked to northward shifts in bird breeding ranges for some species [START_REF] Thomas | Birds extend their ranges northwards[END_REF][START_REF] Hitch | Breeding distributions of North American bird species moving North as a result of climate change[END_REF][START_REF] Gillings | Directionality of recent bird distribution shifts and climate change in Great Britain[END_REF]. If climate change is responsible for declining population sizes at the southern range limit and increasing population sizes at a northern range limit [START_REF] Jiguet | Bird population trends are linearly affected by climate change along species thermal ranges[END_REF], the average latitudinal origin of migrants should move northwards through time. This trend might be detectable as decreasing mean deuterium values in the feathers grown on the breeding grounds. Land-use change is also affecting bird abundances, especially for habitat specialists in the context of biotic homogenization [START_REF] Devictor | Functional biotic homogenization of bird communities in disturbed landscapes[END_REF]. If land use changes occurred heterogeneously in space, especially along latitudinal gradients, they might also provoke latitudinal shifts in relative abundance that could translate into changing δ 2 Hf values.

Naturally occurring patterns of stable isotopes in foodwebs across large geographic areas enable the tracking of bird migration (Hobson and Wassennaar 2019). This is based on the principle that stable isotope ratios in tissues (e.g. feathers, blood) can be correlated with values of food and water consumed by birds following well-described processes [START_REF] Hobson | Making Migratory Connections with Stable Isotopes[END_REF]. Protium ( 1 H) and the heavier isotope, deuterium ( 2 H), occur naturally in environmental waters, and the ratio of the heavier to lighter isotope ( 2 H/ 1 H measured as δ 2 H) in precipitation has been shown to change predictably at continental scales forming "isoscapes" (West et al. 2010). Indeed, δ 2 H values in precipitation (δ 2 Hp) generally have a latitudinal structure across continents [START_REF] Bowen | Global application of stable hydrogen and oxygen isotopes to wildlife forensics[END_REF]) and these patterns form the basis of tracing animal movements because such patterns are transferred up food webs and are locked into metabolically inert tissues such as feathers. Feathers grown at a given location generally have δ 2 H values (δ 2 Hf) proportional to local amount-weighted mean annual or growing-season δ 2 Hp [START_REF] Hobson | Linking Hydrogen (δ 2 H) Isotopes in feathers and precipitation: sources of variance and consequences for assignment to isoscapes[END_REF], so that the geographic origin of migratory birds can be traced by studying stable isotopes in their feathers [START_REF] Hobson | Tracking Animal Migration using Stable Isotopes[END_REF]. In continental Europe, local amountweighted mean δ 2 Hp has a latitudinal structure [START_REF] Bowen | Global application of stable hydrogen and oxygen isotopes to wildlife forensics[END_REF]), so that a potential latitudinal shift in the distribution of a species' abundance could translate into a shift in δ 2 Hf values for feathers grown on the breeding grounds.

Terrestrial isotopic landscapes are presumed to be relatively static across recent time scales providing an opportunity to determine temporal trends in migratory connectivity or trophic networks using stable isotopes (e.g. [START_REF] Hobson | Making Migratory Connections with Stable Isotopes[END_REF][START_REF] Hobson | Migratory connectivity in the Rusty Blackbird (Euphagus carolinus) in North America: Isotopic evidence from feathers of historical and contemporary specimens[END_REF][START_REF] Guillemain | Combining stable isotopes (δ 2 H) with rings and neck collars: geographic origin of Greylag Geese (Anser anser) wintering in Camargue, southern France[END_REF]. Isotopic landscapes are reported to have changed in marine environments [START_REF] Kurle | Spatial and temporal variability within marine isoscapes: implications for interpreting stable isotope data from marine systems[END_REF]McWhorther 2017, de la Vega et al. 2019); however, few studies have investigated temporal trends in terrestrial isoscapes. [START_REF] Duliński | Isotopic composition of precipitation in Poland: a 44-year record[END_REF] found no temporal trend of deuterium in monthly precipitation at Krakow, Poland, over 44 years, though in the same precipitation samples, deuterium concentration decreased with increasing air temperature. Hence, at a continental scale, if a possible shift in the deuterium isoscape could ensure from climate change, such changes have not yet been observed [START_REF] Reckerth | Long-term data set analysis of isotopes in German rivers[END_REF]. Moreover, if occurring, such shifts should be more obvious in recent decades as the effects of climate change have become more pronounced, but there is currently no evidence for such change for deuterium. As such, we retain here the most plausible hypothesis of a static terrestrial deuterium landscape since the mid-19 th century.

Our objective in this study was to compare δ 2 Hf values in samples collected in France over a long time period (from mid 19 th to early 21 st centuries), by considering two migratory passerine species suffering long-term declines as model subjects: Brambling Fringilla montifringilla and Ortolan Bunting Emberiza hortulana. We sought to determine whether historical and modern δ 2 Hf data have similar distributions, or if putative distributional shifts could mirror spatially structured trends in abundance.

Methods

Study species

The Brambling is a short-distance migrant that breeds in the northern forests and taiga with a breeding distribution in the Palearctic, across Scandinavia (except southern Sweden) and Russia towards eastern Siberia. There are no obvious trends reported for the distribution of its breeding range over the last decades. The current winter distribution of this species includes western and southern Europe, from Denmark to Iberia and Turkey. On the wintering grounds, bramblings sometimes form large nocturnal roost sites, where millions of birds gather at night. European breeding populations have declined by 48% since 1980 with a consistent decrease in numbers since the early 1990s (PECBMS 2019) attributed to changes in forest structure and management [START_REF] Lehikoinen | Common montane birds are declining in northern Europe[END_REF] [START_REF] Jiguet | Desert crossing strategies of migrant songbirds vary between and within species[END_REF]. Populations breeding in southern Europe display higher growth rates than those breeding at more northern latitudes [START_REF] Jiguet | An update of European breeding population sizes and trends of the Ortolan Bunting (Emberiza hortulana)[END_REF]. Ortolan Buntings undergo a complete post-breeding moult on the breeding grounds before fall migration, while firstcalendar-year individuals maintain feathers grown in the nest. Hence, migrants captured during the autumn have feathers grown on their breeding grounds.

Sampling Bramblings and Ortolan Buntings

Our Brambling sample first benefited from a collection of single tail feathers collected from individuals caught and ringed at a huge nocturnal roost near Pau (Pyrénées-Atlantiques), France, during 22-31 December 1969 (n=192), preserved at the National Museum of Natural History in Paris, France (MNHN, CRBPO ringing scheme). We also organized sampling of feathers by ringers at their garden feeders across the country in winter 2013 -14 and 2014-15 (n=34), including the collection of feathers at a huge roost in January 2015 (up to 4 million bramblings at Kirchberg, Haut-Rhin, France; n=160). We further sampled museum specimens held at MNHN, collected between 1860 and2009 (n=81). The list of these MNHN specimens is provided in Appendix 1.

For the Ortolan Bunting, we first collected a few body feathers from individuals captured during autumn migration in France in 2012[START_REF] Procházka | Using stable hydrogen isotopes (δ²H) and ring recoveries to trace natal origins in a Eurasian passerine with a migratory divide[END_REF], 2014and 2015 (n=74;[START_REF] Jiguet | Unravelling migration connectivity reveals unsustainable hunting of the declining ortolan bunting[END_REF]. We also collected body feathers from museum specimens collected in France during autumn migration (n=7 for the period 1872-1897; n=7 for 1951-1969). The list of these specimens in provided in Appendix 1. We stored feathers in individual envelopes at ambient temperature until isotopic analysis.

Hydrogen isotopic measurements

Feathers were cleaned in 2:1 chloroform:methanol solvent rinse and prepared for δ 2 H analysis at the Stable Isotope Laboratory of Environment Canada, Saskatoon, Canada. The δ 2 H of the nonexchangeable hydrogen of feathers was determined using the method described by [START_REF] Wassenaar | Comparative equilibration and online technique for determination of non-exchangeable hydrogen of keratins for use in animal migration studies[END_REF] based on two calibrated keratin hydrogen-isotope reference materials (CBS: -197 ‰, KHS: -54.1 ‰). We performed hydrogen isotopic measurements on H2 gas derived from hightemperature (1350 °C) flash pyrolysis (Eurovector 3000; Milan, Italy) of 350 ± 10 ug feather subsamples and keratin standards loaded into silver capsules. Resultant separated H2 was analysed on an interfaced Isoprime (Crewe, UK) continuous-flow isotope-ratio mass spectrometer. Measurement of the two keratin laboratory reference materials corrected for linear instrumental drift were both accurate and precise with typical within-run measurement error < 2 ‰. All results are reported for non-exchangeable H expressed in the typical delta notation, in units of per mil (‰), and normalized on the Vienna Standard Mean Ocean Water-Standard Light Antarctic Precipitation (VSMOW-SLAP) standard scale.

Statistical analyses

To detect potential temporal trends in δ 2 Hf values within species, we performed both Kruskal-Wallis rank sum tests (as some subsamples are of small size), but also generalized linear models, as the specific global datasets did not depart from a normal distribution (Shapiro-Wilk normality tests: bramblings W=0.99, P=0.19; buntings, W=0.97, P=0.054). Potential temporal differences were tested by comparing periods (Kruskal-Wallis tests and linear models) or as a linear temporal trend (linear models). When applicable (for Bramblings only), we also tested for potential differences in δ 2 Hf between age classes and/or sexes in linear models.

Results

Figure 1 shows the boxplot of δ 2 Hf values for Bramblings sampled at different periods (including feathers collected at two dormitories). Average ± s.d. deuterium concentrations for the different groups were: -95. 28 ± 11.00 (1850-1899, n=17), -98.36 ± 10.42 (1900-1949, n=43), -96.38 ± 16.29 (1950-1999, n=19), -94.95 ± 10.37 (2000-2015, n=36), -98.89 ± 13.58 (1969 dormitory), -99.51 ± 12.92 (2015 dormitory).We found no significant difference in δ 2 Hf for the different time periods df=5,P=0.50;linear model,F5,449=1.09,P=0.37), and no difference between sexes (for the subsample of birds of known sex, F1,290=1.23, P=0.27) and between ages (first-winter vs. older, for the subsample of birds of known age, F1,247=1.00, P=0.32). We found no significant temporal linear trend in δ 2 Hf (t453=-0.71, P=0.48) even when excluding the large samples collected in 1969 and 2015 at large roosts (t113=0.62, P=0.54). No temporal trend was detected in the global dataset even after accounting for sex and age (t244=0.60, P=0.55).

Figure 2 shows the boxplot for Ortolan Buntings sampled at three different periods. Average ± s.d. deuterium concentrations were -90.18 ± 19.36 (1872Average ± s.d. deuterium concentrations were -90.18 ± 19.36 ( -1897Average ± s.d. deuterium concentrations were -90.18 ± 19.36 ( , n=7), -75.59 ± 15.32 (1951Average ± s.d. deuterium concentrations were -90.18 ± 19.36 ( -1969Average ± s.d. deuterium concentrations were -90.18 ± 19.36 ( , n=7) and -85.28 ± 15.32 (2012Average ± s.d. deuterium concentrations were -90.18 ± 19.36 ( -2015, n=74), n=74). We found no statistical difference in δ 2 Hf measured from specimens collected during these three periods (Kruskal-Wallis, chi-sq= 1.28, df=2, P=0.53; linear model, F2,85=0.43, P=0.65).

Discussion

We failed to find any temporal structure in the δ 2 Hf values for two species that have declined precipitously over the last century, Brambling and Ortolan Bunting. If climate change was responsible for a substantial northward shift in abundance for these species, we might have detected a decrease in δ 2 Hf values either throughout the studied period, or in more recent periods when temperature increases were more obvious [START_REF] Gregory | An indicator of the impact of climate change on European bird populations[END_REF]. Our failure to detect such a structure might be due to several factors. The first explanation is technical, not biological, as small sample sizes reduce the statistical power to detect any existing trends. Although if this holds for the Ortolan Buntings here, as we had only 7 samples for each of the two old periods, we had large samples of Bramblings. A biological explanation is that population declines occurred similarly at all latitudes across the breeding range, so without provoking an obvious latitudinal shift in abundance. Such a pattern could occur either because the effects of climate change were balanced across latitudes, or because the effects of land-use changes counter-balanced directional effects of climate change. As there is strong support for latitudinal trends structuring long-term population growth rates facing climate change for common songbirds in Europe [START_REF] Jiguet | Bird population trends are linearly affected by climate change along species thermal ranges[END_REF], this would suggest an antagonistic strong latitudinal trend in land-use change, affecting farmland (Ortolan Bunting) as well as woodland (Brambling) habitats. As an example, while [START_REF] Lehikoinen | North by north-west: climate change and directions of density shifts in birds[END_REF] observed a significant northward shift in Brambling abundance in Finland, intensive forestry expanded northward during the 20 th century in the same country [START_REF] Kotilainen | Transformation of forest policy regimes in Finland since the 19th century[END_REF]. Furthermore, the Brambling is a forest-habitat generalist, and therefore its trends are not likely to be explained by habitat alterations caused by forestry [START_REF] Virkkala | Population trends of forest birds in a Finnish Lapland landscape of large habitat blocks: Consequences of stochastic environmental variation or regional habitat alteration?[END_REF], while its range is bounded northwards by the Barents Sea, making any northwards expansion impossible -although an average northwards shift has been found [START_REF] Lehikoinen | North by north-west: climate change and directions of density shifts in birds[END_REF]. We sampled from only one area in France and migratory connectivity between breeding and non-breeding areas for this sampling region may not represent shifts in breeding ranges across the entire populations of both study species. Therefore, similar studies in other parts of the non-breeding range would be useful but are currently unattainable due to a lack of historic samples.

A plausible alternative is that latitudinal shifts occurred but not to the extent of translating into a detectable shift in feather hydrogen isotope values. For example, the northward shift in density of Brambling in Finland since the 1970s is statistically significant but only 64 km, while it is non-significant at 9 km for Ortolan Bunting [START_REF] Lehikoinen | North by north-west: climate change and directions of density shifts in birds[END_REF]. These short and recent shifts in ranges are unlikely to result in a significant shift in δ 2 Hf values because δ 2 Hp gradients occur at larger scales on the order of hundreds of kilometres. Finally, our inferences are based on the assumption that the δ 2 Hp isoscape for Europe has not changed significantly over this period of feather sampling. Unfortunately, the International Atomic Energy Agency (IAEA) Global Network of Isotopes in Precipitation (GNIP) is only ~60 years old. Nonetheless, there appears to be little evidence for a systematic change in the GNIP data for western Europe during that time period based on surface waters [START_REF] Reckerth | Long-term data set analysis of isotopes in German rivers[END_REF]) and the long-term GNIP precipitation isoscape for Europe provides excellent calibration between feather and mean annual δ 2 Hp (Prochazka et al. 2013).

Although there is published evidence for some shifts in marine isoscapes [START_REF] Kurle | Spatial and temporal variability within marine isoscapes: implications for interpreting stable isotope data from marine systems[END_REF][START_REF] De La Vega | Temporal and spatial trends in marine carbon isotopes in the Arctic Ocean and implications for food web studies[END_REF], there is no such evidence for terrestrial environments, while this is also not the general rule in marine environments [START_REF] Mackenzie | Testing the long-term stability of marine isoscapes in shelf seas using jellyfish tissues[END_REF]. Indeed, broad geographic variations in stable carbon and nitrogen isotope distributions across the North Sea are consistent with previously published variations seen in archaeological fish bones from the ninth to seventeenth centuries, despite dramatic changes in land use in the past 1,000 years [START_REF] Mackenzie | Testing the long-term stability of marine isoscapes in shelf seas using jellyfish tissues[END_REF]. Therefore, stable isotopebased studies of movements in animals can consequently be referenced to available modern isoscape models, unless a contradictory evidence is produced. Nevertheless, more retrospective studies are needed for terrestrial ecosystems to definitely validate this assumption [START_REF] Hobson | Migratory connectivity in the Rusty Blackbird (Euphagus carolinus) in North America: Isotopic evidence from feathers of historical and contemporary specimens[END_REF]. 1900-1950 (n=43), 1950-1999 (n=19), 1969 roost (n=192), 2000-2015 (n=24), 2015 roost (n=160). Sample sizes as follows: 1888-1897 (n=7), 1951-1969 (n=7), 2012-2015 (n=74).

  but also to climate change coinciding with a significant 64 km northward shift in average Finnish breeding densities from the 1970s into the 2000s (Lehikoinen and Virkkala 2016). Adult Bramblings have a complete summer moult on their breeding grounds, replacing all flight feathers (including rectrices) in July-September. Firstwinter individuals have at most replaced a few greater coverts (Brown and Mead 2003). Hence, all winter individuals, either adults or first-winter individuals, have feathers grown on the breeding grounds. The Ortolan Bunting is a long-distance Palearctic migrant songbird that breeds across Eurasia from Scandinavia to the Middle East and winters south of the Sahara in Africa. European populations have suffered a continuous decline since 1980, with a 90% reduction in population size during the period 1980-2017 (PECBMS 2019). The main drivers have been identified as habitat destruction through agricultural intensification (Menz and Arlettaz 2012), poaching (Jiguet et al. 2019) and potentially climate change

Figure 1 .

 1 Figure 1. Boxplots showing variation in deuterium values in Brambling feathers (δ 2 Hf in ‰) for different time-periods. Each boxplot reports median, 1 st and 3 rd quartiles, minimum and maximum

Figure 2 .

 2 Figure 2. Boxplot of deuterium concentration values in Ortolan Bunting feathers (δ 2 Hf in ‰) for three different time periods. Each boxplot reports median, 1 st and 3 rd quartiles, min and max values.

Acknowledgements

We are indebted to all ringers who captured, ringed and sampled ortolan buntings and bramblings, under the license provided by the French national ringing scheme CRBPO. FJ thanks Nicolas Minéry for organizing the collection of feathers at the 2015 Kirchberg brambling dormitory. Environment and Climate Change Canada supported the stable isotope analyses through an operating grant to Keith Hobson. We are indebted to the curators of the bird collections at MNHN, at Museum d'Histoire Naturelle de Bayonne, and Museum d'Histoire Naturelle de Toulouse.


Mihoubi Douadi
email: douadi.mihoubi@univ-msila.dz

Lmpa

An approach attack to the strong Goldbach's conjecture

Keywords: 11PXX, 11p32, 11P05, 11B37

This paper is about the famous strong Goldbach's conjecture which states that : every even integer greater than two is a sum of two primes: To investigate in the validity of this amazing statement, we present a new approach attack using the contradiction or the reduction ad absurdum. The idea used here can be summarized as follows: based on the well ordering of the set of natural numbers N , we consider for every even integer 2n 4, with n 2, the …nite sequence of natural numbers S m (n) = (s i (n)) i2f1;2;:::;mg de…ned by: s i (n) = 2n p i , where p i is the i-th prime number in the …nite strictly ordered sequence of primes

where m = (2n) denotes the number of primes p such that p < 2n. We prove in …rst that, if the sequence S m (n) = (s i (n)) i2f1;2;:::;mg do not contains a prime number then the largest terms of the sequence exceeds the number 2n and this is an absurdity because S m (n) is strictly decreasing sequence of natural numbers from 2n 2 to 1, i.e,

Consequently, for any natural number n 2, there exists at least one prime number s r (n) = 2n p r belonging to the sequence S m (n), and we obtain then the expected result 2n = s r (n) + p r , where p r and s r (n) are primes from the set P m . Key Words: Well-ordering (N; <), the product order (N m ; ), basic concepts and theorems on number theory, the indirect and inductive proofs on natural numbers. AMS 2010:

such that every su¢ ciently large even integer is in the set P h + P k of sums of integers of P h and P k . The …rst result in this line of study [START_REF] Brun | le crible d'Eratosthène et le théoreme de Goldbach[END_REF], was obtained by Brun in 1919 by showing that: every su¢ ciently large even number is in P 9 + P 9 . In [START_REF] Rényi | on the representation of even numbers as the sum of a prime and almost prime[END_REF], Rényi proved in 1947 that there exists an integer k 1 such that every su¢ ciently large even integer is in P 1 + P 7 . In 1950, Selberg further improved the result in [START_REF]Selberg The general sieve method and its place in prime number theory[END_REF], by showing that every su¢ ciently large even integers is in P 2 + P 3 . The best result in this direction is due to Chen [START_REF] Chen | On the representation of a large even integer as the sum of a prime and the product of at most two primes, I and II[END_REF] (announcement of results in 1966, proofs in detail in 1973 and 1978) proving that:

Every su¢ ciently large even integer may be written as 2n = p + m, where p is a prime and m 2 P 2 .

The "basis" approach began in 1930 with the theorem of Schnirelmann [START_REF] Schnirelmann | Uber additive eigenschaften von zahlen[END_REF], proving that: There exists a positive integer S, such that every su¢ ciently large even integer is the sum of at most S primes. The best known result currently stems from the proof of the weak Goldbach conjecture by Harald Helfgott [START_REF] Helfgott | The ternary Goldbach conjecture is true[END_REF], which directly implies that every even number n 4 is the sum of at most 4 primes. For further history and progress in the Goldbach problem see, for example, the paper [START_REF] James | Recent progress in the Goldbach Problem[END_REF] and the book [START_REF] Ribenboim | The little Book of Big Primes[END_REF] from respectively R. D. James and P. Ribenboim and the paper "Goldbach's conjecture" from the wikipedia the free encyclopidia [START_REF]Goldbach's conjecture, from Wikipedia, the free encyclopedia[END_REF].

The paper is organised as the follows: The …rst section is devoted to giving a brief history on the Goldbach's conjecture and the idea used in this paper to proving the conjecture. We give in the second section, the theoretical elements from the number theory essential to the paper. The third section, is the main section of the paper, where the sequences S m (n) of natural numbers are given, and we prove that these sequences contains a prime numbers for each natural number n 1. And …nally, in the last subsection 3:2, we deduce the validity of the strong Goldbach's conjecture

The idea used in this paper to prove the conjecture

To prove the conjecture, we consider for every even natural number 2n 4, with n 2, the …nite sequence of natural numbers S m (n) = (s i (n)) i2f1;2;:::;mg de…ned by s i (n) = 2n p i , where p i is the i-th prime number in the …nite strictly ordered sequence of primes P m := p 1 = 2 < p 2 = 3 < p 3 = 5 < ::: < p m where m = (2n) denotes the number of primes p such that p < 2n. To con…rm that, for each natural number n 2, there exists at least one prime number s r (n) = 2n p r belonging to the sequence S m (n), we prove by contradiction that, if each term the sequence S m (n) is a composite number, then it is impossible to write all the terms of S m (n) between the numbers 1 and 2n 2, and this contradict the fact that we have:

2n > s 1 (n) = 2n p 1 = 2n 2 > s 2 (n) = 2n p 2 = 2n 3 > ::: > s i (n) =
2n p i > s i+1 (n) = 2n p i+1 > ::::: > s m (n) = 2n p m 1.

i.e, all the terms of sequence S m (n) are normaly between the natural numbers 1 and 2n.

Consequently, the sequence S m (n) must contains at least one prime s r (n) = 2n p r of P m . We obtain then the long-awaited result 2n = s r (n)+p r , where p r is the r-th prime number of the sequence P m ; This result will con…rms the validity of the Goldbach's statement. It is noted here that any solution on primes p, q of the equation p + q = 2n, for a given natural number n 2, exists if and only if p 2 P m and q = 2n p 2 S m (n) \ P m ; and the smallest of these numbers, suppose that is p, must be between 2 and n, i.e., 2 p n and the other q must be in n q < 2n. For this reason, the sequence S m (n) of natural numbers, as de…ned above, are considered in this paper for proving the conjecture.

Preliminary and theoretical elements essential to the paper

The set of natural numbers N := 1; 2; :::; n; :::, is well ordered using the usual ordering relation denoted by , where any subset of N contains a least element. Another way to see the well-ordering of N is that any natural number n can be reached in …nite counting steps by ascent (adding 1) or descent (subtracting 1) from any other natural number m; there isn't an in…nite descent on natural numbers. This signi…cant characteristic property of the set of naturals numbers N, is the key of almost results of properties of natural numbers.The concept of well-ordering is of fundamental importance in view of the mathematical induction to proving, in two steps only, the validity of a property H (n) depending on natural number n. For the natural numbers a; b, we say a divides b, if there is a natural number q such that b = aq.

In this case, we also say that b is divisible by a, or that a is divisor of b, or that a is a factor of b, or that b is a multiple of a. If a is not a divisor of b, then we write a -b. A natural number p > 1 is called prime if it is not divisible by any natural number other than 1 and p. Another way of saying this is that a natural number p > 1 is a prime if it cannot be written as the product p = t 1 t 2 of two smaller natural numbers t 1 ; t 2 not equal to 1. A natural number b > 1 that is not a prime is called composite. The number 1 is considered neither prime nor composite because the factors of 1 are redundant 1 = 1 1 = 1 1 ::::: 1. We shall denote by p 1 = 2 < p 2 = 3 < p 3 = 5 < p 4 = 7 < ::: < p i < :::

the in…nite increasing sequence of primes, where p i is the i-th prime in this sequence. Euclid's theorem ensures that there are in…nitely many primes, without knowing their pattern and indication of how to determine the i-th prime number. There is no regularity in the distribution of these primes on the chain (N; ); in certain situation they are twins, i.e., there exists a positive integer k such p k+1 = p k + 2, like p 2 = 3 and p 3 = 5, p 5 = 11 and p 6 = 13 (it is not known today whether there are in…nitely many twin primes), while at the same time, for any integer k 2, the sequence of successive k 1 natural numbers k!+2; k!+3; k!+4; :::; k!+k, are all composite, for the simple reason that, any term k! + t, for 2 t k; is divisible by t.

The fundamental theorem of arithmetics shows that any natural number n > 1 can be written as the product of primes uniquely up the order. For the natural number n 2, we denote by (n) the number of primes p n, where (n) is called also the prime counting function, for example (4) = 2,

(5) = 3,...etc.). The fundamental theorem of primes (Tcheybeche¤ gave an empirical estimation around 1850, Hadamard and de Vallée-Poussin theoretical proof at the end of 19 th century) shows that, for any large natural number n, we have (n) n ln n and then p n n ln n where ln denotes the natural logarithm of base e = 2; 71:::. The Bertrand's postulate (1845) provides that between any natural number n 2 and its double 2n there exists at least one prime. Equivalently, this may be stated as (2n) (n) 1, for n 2, or also in compact form: p n+1 < 2p n for n 1. Recall that the notation f (x) h (x) for a positive real valued continuous functions f (x) ; h (x) means that lim x!1 f (x) h(x) = c, with a constant positif c, and in this case f (x) ; h (x) are said to be asymptotically equal as x tends to in…nity. The following result from [START_REF] Bach | Algorithmic number theory, vol I: E¢ cient algorithms[END_REF] is useful for this paper to estimate some results for a large natural number n : The prime sums of the …rst n primes, denoted (n) = m P i=1 p i , is asymptotically equivalent to 1 2 n 2 ln n. For the notions in elementary number theory see for example [START_REF] Stillwell | Numbers and geometry[END_REF] and in Algorithmic number theory see [START_REF] Bach | Algorithmic number theory, vol I: E¢ cient algorithms[END_REF]. Recall that the set of natural N is the disjoint union N = f1g [ Even [Oddc [Odd Pr imes where Even is the set of the even natural numbers, Oddc is the set of odd composite natural numbers and Odd Pr imes is the set of odd primes. Evidentally, the number 2 is the only even prime number.

Recall that a sequence of natural number is an ordered set of numbers that most often follows some rule or pattern to determine the next term in the order. formally, a sequence of real numbers is a mapping u : N ! R, we typically write it as n 7 ! u n . A subsequence of the sequence u is a map v = u where : N ! N is a strictly increasing mapping. Then v n = u ( (n)) . For example, the sequence of primes P is a subsequence of the sequence of natural numbers N, with the respect of the usual ordering . An arithmetic sequence u n+1 = u n + d is in which successive terms di¤er by the same amount d called the reason of the sequence. Dirichlet's theorem on arithmetic progression asserts that if gcd (a; d) = 1 then an + d is prime for in…nitely many natural number n. A series is summation of the terms of a sequence. A …nite series are typically written in the following manner: 3 The sequences S m (n) containing a prime numbers

Before giving the sequences S m (n) used to con…rm the conjecture, we begin by these simple lemmas in view of their usefulness for the rest of the paper.

Lemma 2 Any natural number b 6 = 1 admits a prime divisor. If b is not prime, then there exists a prime p divisor of b such that p 2 b.

Proof. By the de…nition of prime number, if the natural number b 6 = 1 admits only the number b as proper divisor, then b is a prime number. If b is not prime, then it can be factored as b = pq such that: 1 < p < b and 1 < q < b with p is the smallest, under the usual ordering , proper factor of the number b. Since p is the smallest proper factor of b then p must be a prime otherwise, it is not then the least factor of b. As p is the least factor of b then, p q. Multiplying both sides by p, we obtain: pp = p 2 pq = b.

Lemma 3 If the odd integer t > 1 is not prime, then it can be factorised only in the form t = t 1 t 2 where t 1 , t 2 are proper factors 6 = 1, and each factor t 1 and t 2 , it is also an odd natural number greater or equal to the number 3.

Proof. If the odd integer t > 1 is not prime, then it is composite. Let t = t 1 t 2 be any possible factorization of t with t 1 , t 2 are proper factors 6 = 1.

If one of these factors (or both) is an even integer, then the product t 1 t 2 = t will be also an even integer, but the number t is odd. Then each of the factors t 1 and t 2 must be odd and then greater or equal to the number 3.

Let m 1 be a natural number. We denote by I m = f1; 2; :::; mg the set of the …rst m consecutive natural numbers from 1 to m. For any non empty set E, we denote by E m the cartesian product of m copies of E. If m 2, then E m = E m 1 E. An element (a 1 ; a 2 ; :::a m 1 ; a m ) = ((a 1 ; ::

:; a m 1 ) ; a m ) of E m = E m 1 E is called m tuples. It

Proof. By induction on

i 2 I m . For i = 1, if a 1 < b 1 then a 1 < b 1 < b 2 ::: < b m , and we have a 1 = 2 b = (b i ) i2Im , contradiction. For the same reason, if b 1 < a 1 , we obtain b 1 = 2 a = (a i ) i2Im
and then it necessary that a 1 = b 1 . Suppose that we have for some i with m > i

2, a 1 = b 1 , a 2 = b 2 ;...,a i = b i . For i + 1, if a i+1 < b i+1
, then there exists no b j for j i + 1 such that a i+1 = b j and there exists no b j for j i because

a i+1 > a i = b i > a i 1 = b i 1 > ::: > a 1 = b 1 The same argument holds if b i+1 < a i+1 , then it is necessary that we have a i+1 = b i+1 . Consequently, for each i 2 I m : a i = b i .
Note that the condition "a i = b j " for some i; j 2 I m ", in the lemma 3 above, is necessary and su¢ cient condition. It is necessary, because we can exhibit two increasing m sequences

a = (a 1 < a 2 ::: < a m ), b = (b 1 < b 2 < ::: < b m ) with m P i=1 a i = m P i=1 b i and a i 6 = b i for each i 2 I m . For example, a = (2 < 10)
and b = (4 < 8).

The sequences S m (n)

Let n 2 be a natural number. We consider the …nite strictly increasing sequence of the …rst m consecutif prime numbers P m : p 1 = 2 < p 2 = 3 < p 3 = 5 < ::: < p i ::: < p m where m = (2n) denotes the number of primes p < 2n. Let P m = (p i ) i2Im denotes the …nite successive primes strictly smaller than 2n. The Bertrand postulat asserts that at least the prime p m is between n and its double 2n. For any natural number n > 2, we consider the …nite sequence S m (n) = (s i (n)) i2Im of natural numbers de…ned by: s i (n) = 2n p i , where p i is the i-th prime of P m . Then we have:

s 1 (n) = 2n 2, s 2 (n) = 2n 3, s 3 (n) = 2n 5, . . . s i (n) = 2n p i , . . . s m (n) = 2n p m .
Example 5 For n = 10, the …nite sequence of primes less than 20 are:

p 1 = 2 < p 2 = 3 < p 3 =
(n) = (s i (n)) i2Im de…ned by s i (n) = 2n p i , with 1 i m, is strictly decreasing from s 1 (n) = 2n 2 = max (S m (n)) to s m (n) = 2n p m = min (S m (n))
1, and each element s i (n) of this sequence is an odd natural number except the …rst term s 1 (n) = 2n 2 that is evidently an even number. The last term s m (n) is equal to 1 only in the case when p m = 2n 1.

Proof. Let n > 2 be a natural number with m = (2n). Since the …nite sequence of primes P m : p 1 = 2 < p 2 = 3 < p 3 = 5 < ::: < p i ::: < p m is strictly increasing, and each term s i (n) is de…ned by 2n p i , then the sequence S m (n) is strictly decreasing from s 1 (n) to s m (n). In fact, we have p i+1 > p i and then s i (n) = 2n p i > s i+1 (n) = 2n p i+1 for each i with 1 i m 1. this shows that we have:

s 1 (n) = 2n p 1 = 2n 2 > s 2 (n) = 2n p 2 = 2n 3, s 2 (n) > s 3 (n) = 2n p 3 = 2n 5, ., ., ., s i (n) = 2n p i > s i+1 (n) = 2n p i+1 > ::::: > s m (n) = 2n p m 1.
Since for each i, with 2 i m, the prime p i is odd, then the term s i (n) = 2n p i is also an odd natural number. The …rst term s 1 (n) = 2n 2, is the unique even number in the sequence S m (n). The last term s m (n) = 2n p m = min (S m (n)) is equal to the number 1 if and only if p m = 2n 1. In fact, if p m = 2n 1 then s m (n) = 2n p m = 2n (2n 1) = 1. In the reverse case, we have p < 2n, 8p 2 P m , and then 2n p > 0 () 2n p 1; and we have 2n p = 1 only in the case when p = 2n 1 = p m . In the example 4, we have this situation, as p 8 = 19 then s 8 (10) = 20 19 = 1. The strict ordering on N m noted can be also de…ned by: a = (a 1 ; a 2; :::; a m ) b = (b 1 ; b 2; ::

:; b m ) if a i < b i for each i 2 I m .
Evidently, the ordering on N m is not a total ordering, for example the sequences (1; 2; 3), (1; 1; 4) of N 3 are not comparable. But (N m ; ) contains the least element (1; 1; :::; 1) (m terms of 1), which satis…es (1; 1; :::; 1) a = (a 1 ; a 2; :::; a m ) for every a 2 N m Notice that for each integer m 1, the set N m equipped the operation of addition of m-sequences de…ned by : For every two m tuples a = (a 1 ; a and this element of N 7 is the smallest element among the set of all the 7-tuples of odd composite natural numbers written in increasing order. Lemma 9 Let (c 1 ; c 2 ; c 3 ; :::; c m 1 ; c m ) be the smallest element of Oddci (m). If m odd composite natural numbers fa 1 ; a 2 ; :::; a m g are such that a 1 < a 2 < ::: < a m with a 1 c 1 then a i c i , for each i with 1 i m. And consequently we have

m P i=1 a i m P i=1 c i .
Proof. The proof is by induction on i 2 I m . For i = 1 we have a 1 c 1 , and then the hypothesis is given true for i = 1 . For i = 2, we have a 2 > a 1 c 1 then a 2 c 2 because c 2 is the …rst odd composite after c 1 i.e., c 2 is the consecutive odd composite of c 1 , and then the hypothesis is true for i = 2 . Suppose that for some i 2, we have a i c i . Since a i+1 > a i and a i c i then a i+1 > a i c i , and since the …rst odd composite after a i is a i+1 then a i+1 c i+1 . We conclude by induction on i that we have, for each i 2 I m , a i c i . Consequently, we have

m P i=1 a i m P i=1 c i .
Unfortunately, we haven't an explicit formula which determines the m-th odd composite natural number, but we can proceed as follows to enumerate them in increasing order. Recall that, the odd composite numbers are the odd positive integers of the form a (i; j) = (2i + 1) (2j + 1) = 4ij + 2 (i + j) + 1 where (i; j) 2 N 2 with 1 i j (because a is a symetric function on i and j).

Equivalently, an odd composite numbers are the integers of the form a (p; s) = 2 (2p + s) + 1 where p = ij and s = i + j with (i; j) 2 N 2 and 1 i j. Then, we can enumerate these numbers according the increasing values of the product p and then the increasing correspondant sum s. In other words for each p, from p = 1; 2; 3; :::, we compute the couple (i; j), solution for the equation p = ij, with the condition 1 i j, and then the correspondant sum s = i + j, and …nally the resulting odd composite number a (p; s). For example, For p = ij = 1 the unique solution is (i; j) = (1; 1) and then s = 2 which gives a 1 = 2 (2 1 + 2) + 1 = 9. For p = ij = 2 the unique solution is (i; j) = (1; 2) and then s = 3 which gives a 2 = 2 (2 2 + 3) + 1 = 15. For p = ij = 3 the unique solution is (i; j) = (1; 3) and then s = 4 which gives a 3 = 2 (2 3 + 4) + 1 = 21. For p = ij = 4 the …rst solution is (i; j) = (2; 2) and then s = 4 which gives a 4 = 2 (2 4 + 4) + 1 = 25. and the second solution is (i; j) = (1; 4) and then s = 5 which gives a 5 = 2 (2 4 + 5) + 1 = 27,..., and so on. Since we haven't a closed formula which determines the m-th odd composite natural number in function of m, we cannot also write, in function of the parametre m, all the terms of the smallest m tuple i.e, T 2m C m T 3m . Or equivalently, in terms of …nite serie, P T 2m < P C m < P T 3m and we have T 3m is an element of Oddci (m) but T 2m is not in Oddci (m). Evidently, we have T 2m T 3m .

It is noted here that the smallest element C of the set Oddci (without a restricted m) contains in increasing order: each element of the sequence T 3 = (3 (2k + 1)) k 1 , and also each element from the sequence T 5 = (5 (2k + 1)) k 2 which are not multiple of the number 3 and each element from the sequence T 7 = (7 (2k + 1)) k 3 which are not multiple of the number 3 or the number 5, ... and each element from the sequence T p i = (p i (2k + 1)) 2k+1 p i , where p i is the i-th prime, which are not multiple of the primes p 1 or p 2 or ...or p i 1 ... and so on. This process of the construction of the smallest element C shows that each element of the sequence T 3 = (3 (2k + 1)) k 1 is also an element of C, i.e, T 3 is a subsequence of the sequence C and the …rst element of C is the …rst element of the sequence T 3 . And since the di¤erence between two consecutive elements of T 3 is exactly 6, then each other element of C, taken from the others sequence T 5 or T 7 or...or T p i ..., is inserted in increasing order between two appropriate consecutif elements of T 3 . Then the gaps d i = c i+1 c i , for each i 2 N, between two consecutive odd composite natural numbers cannot exceed the number 6, and we have d i 2 f2; 4; 6g because the di¤erence between any two odd natural numbers is even, then 2 d i 6. For example, the …rst four gaps are: The above remarks, shows that we have the following result.

d 1 = c 2 c 1 =
Lemma 10 For every natural number m 4, the smallest element Notice that the above result is equivalent to say that the sequence C m is limited between the sequence m = (9 + 2i) i2f0;1;:::;m 1g and the sequence T 3m = (3 (2i + 1)) i2f1;:::;mg , i.e., m C m T 3m in the product order (N m ; ).

C m = (c 1 < c 2 < ::: < c m ) of the subset Oddci (m), is limited between m C m T 3m ,
Example 11 For example, the …rst ten successive odd composite numbers

9 +6 ! 15 +6 ! 21 +4 ! 25 +2 ! 27 +6 ! 33 +6 ! 35 +6 ! 39 +6 ! 45 +4 ! 49
are limited in the top by the …rst ten odd composite numbers multiple of 3 :

9 +6 ! 15 +6 ! 21 +6 ! 27 +6 ! 33 +6 ! 35 +6 ! 39 +6 ! 45 6 ! 51 +6 ! 57
or equivalently in the product order (N 10 ; ), we have: and this example shows that the sequence of the multiple of 3 grow faster than the sequence the odd composite natural numbers since the sequence of the multiple of 3 is a sub-sequence of the sequence of odd composite natural numbers.

(
The following lemmas gives some lower bounds of the m-th odd composite natural number c m .

Lemma 12

The m-th odd composite natural number c m is greater than 2m with m = (2n).

Proof. The proof is by induction on the natural m 1. For m = 1, we have and c 1 = 9 > 2 1, consequently, the hypothesis is true for m = 1. Suppose that for some natural m 1, we have c m > 2m with m = (2n). For m + 1, we have the gabs d = c m+1 c m 2 f2; 4; 6g, and then c m+1 c m 2, or equivalently c m+1 c m + 2. Since c m > 2m by the hypothesis for m 1, we obtain then c m+1 c m + 2 > 2m + 2 = 2 (m + 1), consequently c m+1 > 2 (m + 1). We conclude by induction on m that we have c m > 2m, for each m 1.

Even better, we have: Lemma 13 For each natural n 1, we have c m > 2n with m = (2n) and c m is the m-th odd composite number.

Proof. The proof is by induction on the natural n 1. For n = 1, we have m = (2 1) = 1 and c 1 = 9 > 2 1 = 2, consequently, the hypothesis is true for n = 1. For n = 2, we have m = (4) = 2. And the assertion c 2 = 15 > 2 2 = 4 is also true for n = 2. Suppose that for some natural n 1, we have c m > 2n with m = (2n). For n + 1, we have m 0 = (2 (n + 1)) = m + 1 in the case when p m+1 = 2n + 1, and (2 (n + 1)) = m in the case when p m+1 6 = 2n+1. In the case p m+1 = 2n+1 then c m+1 > 2n+1 since c m > 2n by the hypothesis for n. And in the case p m+1 6 = 2n + 1, we have also c m+1 > 2n + 1 since c m is at least equal to 2n + 1. In either cases, we have c m+1 > 2 (n + 1) since d = c m+1 c m 2 f2; 4; 6g. We conclude by induction on n that we have, for each n 1, c m > 2n with m = (2n).

Given the importance of above lemma for the paper, we give an other proof by contradiction and we con…rm the validity of the lemma for a large n. Proof. The proof is by contradiction. Suppose that, in the contrary, we have for certain n 1, c m 2n. Then, the interval f1; 2; ::::; 2ng contains m primes + m composites +n even numbers. Since the di¤erence between two consecutif odd numbers is 2. And last odd number in the interval is 2n 1, which it is a prime or an odd composite number, in either cases we have then : 1 + 2m = 2n 1 =) m = n 1: Then we have (n 1) + (n 1) + n = 2n; and this is an absurdity. Consequently, we have for each n 1; c m > 2n.. For a large n;we can con…rm also that we have the following:

The fundamental theorem of primes lim n log n=p n = 1 is equivalent to: the n th prime p n is about n log n , for large n. And this may be interpreted as asserting that, if there are n primes in an interval, then the length of the interval is roughly n log n. Since we have m primes, with m = (2n) , then 2n m log m Suppose that c m 2n., then we have m + m = n (m primes+m composites= n odd numbers between 1 and 2n). Then 2m = n =) 2m = (m log m) =2 and this implies that log m=4 which is absurd for a large n. The following lemma asserts that, if each terms of the sequence S m (n) = (s i (n)) i2Im , is a composite natural number, then the largest term of the sequence will exceed the number 2n, and this will contradict the fact that the greatest element of the sequence S m (n) is equal to 2n 2.

Lemma 14 If each term of the sequence S m (n) is a composite natural number then at least the maximal element of this sequence will be exceeds the number 2n.

Proof. From the lemma 6, all the terms of sequence S m (n) are an odd numbers except the …rst s 1 (n) = (2n 2). Since each term s i (n) 6 = 1 it is supposed to be a composite number, we consider then the possible factorization of each odd term s i (n) as the following form:

s i (n) = p 0 i (n) q i (n)
such that p 0 i (n) is the smallest prime number dividing s i (n), (the existence of this prime factor is assured by the fundamental theorem of arithmetic or it su¢ ces to see the Lemma 2, and q i (n) is the other proper factor). According to the lemmas 3 and 6, the factors p 0 i (n), q i (n) are odd 3, for all i 2 f2; :::; mg in the case when p m 6 = 2n 1, and for all i 2 f2; :::; m 1g in the case when p m = 2n 1 (because in this case, we have s m = 2n p m = 2n (2n 1) = 1). The term s 1 (n) = 2n 2 = 2(n 1) is the only natural even number of the sequence S m (n), which it is evidently a composite number.

Since s i (n) = 2n p i > 2n p i+1 = s i+1 (n), we have also,

s i (n) = p 0 i (n) q i (n) > p 0 i+1 (n) q i+1 (n) = s i+1 (n), for each i 2 f1; :::; m 1g. Consequently, s 1 (n) = p 0 1 (n) q 1 (n) > s 2 (n) = p 0 2 (n) q 2 (n) > ::: > s m (n) = p 0 m (n) q m (n) 1.
Two cases will be considered, depending on whether p m 6 = 2n 1 (in this case s m (n) = 2n p m 3 it is also a composite number) or, p m = 2n 1 (in this case s m (n) = 2n p m = 1 it is neither prime nor a composite number).

1 st case: if p m 6 = 2n 1, then each term s i (n) is to be considered a composite number, and we have:

s 1 (n) = 2n p 1 = p 0 1 (n) q 1 (n) with p 0 1 (n) 2 and q 1 (n) 3, then s 1 (n) 9, s 2 (n) = 2n p 2 = p 0 2 (n) q 2 (n) with p 0 2 (n) 3 and q 2 (n) 3, then s 2 (n) 9, ., ., ., s m (n) = 2n p m = p 0 m (n) q m (n) with p 0 m (n) 3 and q m (n) 3, then s m (n) 9. But, since s 1 (n) > s 2 (n) > :::: > s m (n) c 1 = 9,
is strictly decreasing sequence of odd composite numbers, then certainely, s m 1 (n) c 2 = 15, s m 2 (n) c 3 = 21, ..., and s 1 (n) c m (the m-th odd composite number). According with the lemma 13 we have the c m > 2n. Consequently, we have s 1 (n) c m > 2n. For the second case, i.e., if p m = 2n 1 then s m (n) = 2n p m = 2n (2n 1) = 1, and in this case we have:

s 1 (n) > s 2 (n) > :::: > s m 1 (n) 9 = c 1 , and consequently, s 1 (n) c m 1 .
But we have c m 1 2 (n 1)(from the lemma 13) and since c m 1 is an odd composite and the prime p m is equal to 2n 1, in this case, certainly c m 1 > 2n + 1 and then s 1 (n) c m 1 > 2n + 1. Consequently, from the cases 1 and 2 we have s 1 (n) > 2n , if each term of the sequence S m (n) is to be considered a composite natural number.

Existence of prime in the sequence

S m (n) = (s i (n)) i2I m ,
for any natural number n 2 with m = (2n)

Theorem 15 For any natural number n 2 with m = (2n), the …nite sequence of natural numbers S m (n) = (s i (n)) i2Im de…ned by s i (n) = 2n p i , with 1 i m, contains at least one prime s r 2 P m \ S m (n).

Proof. For each natural number n 2, let S m (n) = (s i (n)) i2Im be the …nite sequence of natural numbers as de…ned in the section 3 above with m = (2n). The proof is by contradiction, and so we begin by assuming that the following hypothesis H (n 0 ) is true for some natural number n 0 2.

The hypothesis H (n 0 ):

"There exists a natural number n 0 2, such that each term s i (n 0 ) 2 S m (n 0 ), for each i 2 I m , is not a prime number". This is equivalent to say : "there exists a natural number n 0 2, such that each term s i (n 0 ) 2 S m (n 0 ), for each i 2 I m , is a composite number or equal to the natural number 1". Symbolically the hypothsis H (n 0 ) can be written as "9 (n 0 > 2) 2 N; 8 i 2 f1; 2; 3; :::mg: the term s i (n 0 ) is not a prime number" We recall that, the unique term s i (n 0 ) of S m (n 0 ), which can be equal to the number 1 is the last term s m (n 0 ) = 2n 0 p m , in the case when p m = 2n 0 1 (see Lemma 6). The last term s m (n 0 ) = 2n 0 p m is the unique term of the sequence S m (n 0 ), which is neither prime nor a composite number in the case when p m = 2n 0 1, i.e., in the case when s m (n 0 ) = 1.

To contradict or reject the hypothesis H (n 0 ) for all n 0 > 2, (in symbolic terms this contradiction is written: 8 (n > 2) 2 N; 9i 2 f1; 2; 3; :::mg such that s i (n) is a prime number). Using the lemma 13, if the hypothesis H (n 0 ) is true for some natural number n 0 2, then s 1 (n 0 ) = max S m (n 0 ) c m > 2n 0 and this is an absurdity because s 1 (n 0 ) = 2n 0 2 = max S m (n 0 ) < 2n 0 and s m (n 0 ) = 2n 0 p m 1. Consequently, it is impossible to write all the elements of sequence S m (n 0 ) = (s 1 (n 0 ) > s 2 (n 0 ) > ::: > s m (n 0 )) between the numbers 1 and 2n 0 2 if each term of sequence it is assumed to be an odd composite number except the …rst term, which is evidently an even composite term. Consequently, the refutation of the argument H (n 0 ) for all n 0 > 2. Then, it is impossible to write each term of sequence S m (n) = (s i (n)) i2Im as an odd composite number between the numbers 1 and 2n 0 2, if each term of this sequence it is assumed to be a composite natural number, and consequently, there must be at least one term s r (n) = 2n p r = p 0 r (n) q r (n), which is not a composite number, then necessarily that the factor q r (n) is strictly inferior to the number 3, and because q r (n) is an odd number, it can not be equal to the number 2 and then we have certainely q r (n) = 1 and then s r (n) = 2n p r = p 0 r (n) q r (n) = p 0 r (n) 1, i.e, the prime p 0 r (n) is equal to the term s r (n) = 2n p r = p 0 r (n) with p 0 r (n) 2 P m ( where P m is the set of primes inferior strictly to the number 2n). Therefore, for each (n 2) 2 N, the sequence S m (n) = (s i (n)) i2Im contains at least one term s r (n), with r 2 I m , which it is a prime number, i.e, s r (n) 2 P m \ S m (n).

Theorem 16 Every even integer 2n 4, with n 2, is a sum of two primes.

20

Proof. If n = 2 then, 4 = 2 + 2. If n > 2, we consider the …nite sequence of primes P m := (p 1 = 2 < p 2 = 3 < :::: < p m ) with m = (2n) and let S m (n) = (s i (n)) i2Im be the …nite sequence of natural numbers de…ned by s i (n) = 2n p i , where p i is i-th prime of P m . From the theorem 15, there exists at least one prime number s r (n) 2 S m (n). Since we have s r (n) = 2n p r , with p r is the r-th prime number of sequence P m , we obtain the result 2n = p r + s r (n). It follows that the Goldbach's conjecture is e¤ectively a theorem of number theory.

As a consequence of this result, given an even natural number 2n 4 with n 2, to …nd the pair of primes numbers (p; s) such that 2n = p+s, it su¢ ces that the algorithm runs through the …nite sequence S m (n) = (s i (n)) i2Im , which contains, at least one solution of this equation for each natural number n 2.

  Suppose we have the following arithmetic serie fa + (a + d) + (a + 2d) + ::: + a + (m 1) dg then

  kd) = m (a + (a + (m 1) d)) =2.

Lemma 4

 4 follows that two ordered m tuples (a 1 ; a 2 ; :::; a m ), (b 1 ; b 2 ; :::; b m ) are equal if and only if a 1 = b 1 ; :::; a m = b m . Let m 1 be a natural number. Let a = (a 1 < a 2 ::: < a m ), b = (b 1 < b 2 < ::: < b m ) be two strictly an increasing m-sequences of natural numbers such that for each i 2 I m there exists j 2 I m such that a i = b j . Then we have for each i 2 I m : a i = b i .

Lemma 6

 6 5 < p 4 = 7 < p 5 = 11 < p 6 = 13 < p 7 = 17 < p 8 = 19. Consequently, (20) = 8 and then the sequence S 8 (n) = S 8 (10) = (s i (10)) i2f1;2;:::;8g is : s 1 (10) = 20 2 = 18, s 2 (10) = 20 3 = 17, s 3 (10) = 20 5 = 15, s 4 (10) = 20 7 = 13, s 5 (10) = 20 11 = 9, s 6 (10) = 20 13 = 7, s 7 (10) = 20 17 = 3, s 8 (10) = 20 19 = 1. For the natural number n 2 with m = (2n), the …nite sequence of natural numbers S m

3. 1

 1 The smallest m-tuples of the set of the strictly increasing m-sequences of odd composite natural numbersWe denote by Odd = f1; 3; 5; :::g = f2k + 1=k 2 N [ f0gg the set of odd natural numbers and by Oddc the set of odd composite natural numbers, i.e., Oddc = fa 2 N=a = (2i + 1) (2j + 1) with i; j 2 N and 1 i jg = f9; 15; 21; 25; :::g.Evidently, we have Oddc Odd N.Let N m = N N ::: N be the cartesian product of m copies of the set of natural numbers N. The usual ordering on N can be extended to the ordering on N m , called the product order, in the following natural way: For every two m tuples a = (a 1 ; a 2; :::; a m ), b = (b 1 ; b 2; :::; b m ) of N m , we de…ne a = (a 1 ; a 2; :::; a m ) b = (b 1 ; b 2; :::; b m ) if a i b i for each i 2 I m .

Lemma 7 Example 8

 78 2; :::; a m ), b = (b 1 ; b 2; :::; b m ) of N m , then a + b = (a 1 + b 1 ; a 2 + b 2 :::; a m + b m ) 2 N m is a semigroup with the neutral element 0 m = (0; 0; :::; 0) (m-zeros).An important subset of N m , which concern this paper, is the subset of N m of the strictly increasing m tuples (or m sequences) of m odd composite natural numbers, which we denote, for the brevity, by Oddci (m). Then, the set Oddci (m) is de…ned by: Oddci (m) = 8 < : a = (a 1 < a 2 < ::: < a m ) 2 N m = a i is an odd composite natural number, for each i 2 I m and a i < a i+1 for every 1 i m 19 =; Evidently, we have Oddci (m) (Oddc) m N m . Since we have Oddci (m) N m , then the subset Oddci (m) is also partially ordered with the induced ordering of N m . For example, the 4-tuples a = (9 < 15 < 21 < 25), b = (21 < 35 < 39 < 63) 2 Oddci (4) with a b. In this paper, the phrases "increasing m-sequence" and "increasing m-tuples" have the same signi…cance and are used equally. For further notions of ordering see[START_REF] Hrbacek | Introduction to set theory[END_REF]. For any natural number m 1, the set Oddci (m) N m contains a smallest element.. Proof. Based on the well ordering of the set N, the smallest elementC m = min Oddci (m) = (c 1 < c 2 < ::: < c m 1 < c m )of the set Oddci (m) is equal to the m tuples of the …rst m consecutif odd composite natural numbers, which can be computed recursively from i = 1 to i = m, in the following manner: For i = 1, the smallest element of the set Oddc is the number c 1 = min fOddcg = min f9; 15; 21; 25; :::; g = 9, and then C 1 = (c 1 ) =[START_REF] Schnirelmann | Uber additive eigenschaften von zahlen[END_REF]. For i = 2, The second element c 2 of set Oddc is the number c 2 = 15 = min fOddc f9gg, consequently, we have C 2 = ((C 1 ) ; c 2 ) = ((C 1 ) ; 15) = ((c 1 ) ; 15) = (9; 15). Suppose that for i = m 1, we have: C m 1 = min Oddci (m 1) = (c 1 ; c 2 ; c 3 ; :::; c m 1 ) = (9; 15; 21; :::; c m 1 ), with c m 1 is the (m 1)-th odd composite natural number of the set Oddc then, the m-th odd composite natural number of the set Oddc is c m = min fOddc fc 1 ; c 2 ; c 3 ; :::; c m 1 gg, and consequently we obtain: C m = (C m 1 ; c m ) = ((c 1 ; c 2 ; c 3 ; :::; c m 1 ) ; c m ) = (c 1 ; c 2 ; c 3 ; :::; c m 1 ; c m ) For example, for m = 7 we have: min Oddci (7) = (9 < 15 < 21 < 25 < 27 < 33 < 35)

C

  m = (9; 15; 21; 25; :::; c m ) = (3 3; 3 5; 3 7; 5 5; :::; c m ), but we can assert in the following lemma 7, for each m 1, the smallest element C m is between the m tuple T 2m = (2 3; 2 5; :::; 2 (2m + 1)) = 2 (3; 5; :::; (2m + 1)) 2 N m and the m tuple T 3m = (3 3; 3 5; :::; 3 (2m + 1)) = 3 (3; 5; :::; (2m + 1)) 2 N m

2 N

 2 15 9 = 6, d 2 = c 3 c 2 = 21 15 = 6, d 3 = c 4 c 3 = 25 21 = 4, d 4 = c 5 c 4 = 27 25 = 2.In general manner, we have: c 1 = 9; and c i+1 = c i + d i , with d i 2 f2; 4; 6g, for each i 2 N. The …rst m consecutive odd composite natural numbers satisfy the following pseudo-recurrence equations:c 1 = 9, c 2 = c 1 + d 1 = 9 + d 1 , with d 1 2 f2; 4; 6g, but of cours d 1 = 6 in this case. c 3 = c 2 + d 2 = 9 + d 1 + d 2 ,with d 1 , d 2 2 f2; 4; 6g, but we have exactly d 1 = 6 and d 2 = 6 in this case. . . . c m = c m 1 + d m 1 = c m 2 + d m 2 + d m 1 = 9 + d 1 + d 2 + ::: + d m 1 , with d i 2 f2; 4; 6g for each i 2 f1; 2; :::; m 1g. We conclude that: for each m 2 N, the m-th odd composite natural c m is limited between the natural numbers N m = 9 + (2 + 2 + ::: + 2) = 9 + 2 (m 1) (the sum of 2 is taken (m 1) times), and the natural number N 0 m = 9 + (6 + 6 + ::: + 6) = 9 + 6 (m 1) i.e., 9 + 2 (m 1) c m 9 + 6 (m 1) for m Since the …rst four gaps are exactly d 1 = d 2 = 6, d 3 = 4 and d 4 = 2, we obtain then the strict inequality : for any natural number m 4. 9 + 2 (m 1) < c m < 9 + 6 (m 1)

  where m = (9 + 2i) i2f0;1;:::;m 1g and T 3m = (3 (2i + 1)) i2f1;:::;mg . Or equivalently, the sum m P i=1 c i of the …rst m consecutif odd composite natural numbers is limited between m (m + 8) < m P i=1 c i < 3m (m + 2). Proof. The sequence C m = (c 1 < c 2 <; :::; < c m ) of the …rst m consecutif odd composite natural numbers can be de…ned recursively by: c 1 = 9 and c i+1 = c i + d i , for each i 2 f1; 2; :::; m 1g, with d i 2 f2; 4; 6g. Then: m P i=1 c i = c 1 + c 2 + ::: + c m = 9 + (9 + d 1 ) + ::: + (9 + d 1 + ::: + d m 1 ). Since for each 1 i m 1, we have d i 2 f2; 4; 6g then m 2i) = m (m + 8) is a lower integer bound of m

Acknowledgement 17 This work was supported by the LMPA, University of M'sila, Algeria.