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A New Proof for Congruent Number’s Problem via Pythagorician Divisors

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Advances in Pure Mathematics 2024年第4期14卷 283-302页

作者： Léopold Dèkpassi Keuméan françois Emmanuel Tanoé UFR Mathematics and Computer Science F&#233lix Houphou&#235t Boigny University Abidjan Cote d&#8217Ivoire

Considering Pythagorician divisors theory which leads to a new parameterization, for Pythagorician triplets ( a,b,c )∈ ℕ 3∗ , we give a new proof of the well-known problem of these particular squareless numbers n∈ ℕ ∗ , called congruent numbers, characterized by the fact that there exists a right-angled triangle with rational sides: ( A α ) 2 + ( B β ) 2 = ( C γ ) 2 , such that its area Δ= 1 2 A α B β =n;or in an equivalent way, to that of the existence of numbers U 2 , V 2 , W 2 ∈ ℚ 2∗ that are in an arithmetic progression of reason n;Problem equivalent to the existence of: ( a,b,c )∈ ℕ 3∗ prime in pairs, and f∈ ℕ ∗ , such that: ( a−b 2f ) 2 , ( c 2f ) 2 , ( a+b 2f ) 2 are in an arithmetic progression of reason n;And this problem is also equivalent to that of the existence of a non-trivial primitive integer right-angled triangle: a 2 + b 2 = c 2 , such that its area Δ= 1 2 ab=n f 2 , where f∈ ℕ ∗ , and this last equation can be written as follows, when using Pythagorician divisors: (1) Δ= 1 2 ab= 2 S−1 d e ¯ ( d+ 2 S−1 e ¯ )( d+ 2 S e ¯ )=n f 2;Where ( d, e ¯ )∈ ( 2ℕ+1 ) 2 such that gcd( d, e ¯ )=1 and S∈ ℕ ∗ , where 2 S−1 , d, e ¯ , d+ 2 S−1 e ¯ , d+ 2 S e ¯ , are pairwise prime quantities (these parameters are coming from Pythagorician divisors). When n=1 , it is the case of the famous impossible problem of the integer right-angled triangle area to be a square, solved by Fermat at his time, by his famous method of infinite descent. We propose in this article a new direct proof for the numbers n=1 (resp. n=2 ) to be non-congruent numbers, based on an particular induction method of resolution of Equation (1) (note that this method is efficient too for general case of prime numbers n=p≡a ( ( mod8 ) , gcd( a,8 )=1 ). To prove it, we use a classical proof by induction on k , that shows the non-solvability property of any of the following systems ( t=0 , corresponding to case n=1 (resp. t=1 , corresponding to case n=2 )): ( Ξ t,k ){ X 2 + 2 t ( 2 k Y ) 2 = Z 2 X 2 + 2

关键词： Prime Numbers-Diophantine Equations of Degree 2 & 4 Factorization Greater Common Divisor Pythagoras Equation Pythagorician Triplets Congruent Numbers Inductive Demonstration Method Infinite Descent BSD Conjecture

New Asymptotic Results on Fermat-Wiles Theorem

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Advances in Pure Mathematics 2024年第6期14卷 421-441页

作者： Kimou Kouadio Prosper Kouakou Kouassi Vincent Tanoé françois UMRI MSN Felix Houphouet-Boigny National Polytechnic Institute Yamoussoukro Ivory Coast Nangui Abrogoua University Applied Fundamental Sciences Department Abidjan Ivory Coast UFRMI Flix Houphout-Boigny University Abidjan Ivory Coast

We analyse the Diophantine equation of Fermat xp yp = zp with p > 2 a prime, x, y, z positive nonzero integers. We consider the hypothetical solution (a, b, c) of previous equation. We use Fermat main divisors, Diophantine remainders of (a, b, c), an asymptotic approach based on Balzano Weierstrass Analysis Theorem as tools. We construct convergent infinite sequences and establish asymptotic results including the following surprising one. If z y = 1 then there exists a tight bound N such that, for all prime exponents p > N , we have xp yp zp.

关键词： Fermat’s Last Theorem Fermat-Wiles Theorem Kimou’s Divisors Diophantine Quotient Diophantine Remainders Balzano Weierstrass Analysis Theorem

Fermat and Pythagoras Divisors for a New Explicit Proof of Fermat’s Theorem:a⁴ + b⁴ = c⁴. Part I

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Advances in Pure Mathematics 2024年第4期14卷 303-319页

作者： Prosper Kouadio Kimou françois Emmanuel Tanoé Kouassi Vincent Kouakou Laboratoire d’ Informatique et de Math&#233matiques appliqu&#233es Institut Polytechnique F&#233lix Houphou&#235t BOIGNY Yamoussoukro Cote d&#8217Ivoire UFR Mathé matiques et Informatique Universit&#233 F&#233lix Houphouet BOIGNY Abidjan Cote d&#8217Ivoire UFR Sciences Fondamentales Appliqué es Universit&#233 NANGUI ABROGOUA Abidjan Cote d&#8217Ivoire

In this paper we prove in a new way, the well known result, that Fermat’s equation a4 + b4 = c4, is not solvable in ℕ , when abc≠0 . To show this result, it suffices to prove that: ( F 0 ): a 1 4 + ( 2 s b 1 ) 4 = c 1 4 , is not solvable in ℕ , (where a 1 , b 1 , c 1 ∈2ℕ+1 , pairwise primes, with necessarly 2≤s∈ℕ ). The key idea of our proof is to show that if (F0) holds, then there exist α 2 , β 2 , γ 2 ∈2ℕ+1 , such that ( F 1 ): α 2 4 + ( 2 s−1 β 2 ) 4 = γ 2 4 , holds too. From where, one conclude that it is not possible, because if we choose the quantity 2 ≤ s, as minimal in value among all the solutions of ( F 0 ) , then ( α 2 ,2 s−1 β 2 , γ 2 ) is also a solution of Fermat’s type, but with 2≤s−10) is solvable in ( a 1 ,2 s b 1 , c 1 ) , s ≥ 2 like above;afterwards, proceeding with “Pythagorician divisors”, we creat the notions of “Fermat’s b-absolute divisors”: ( d b , d ′ b ) which it uses hereafter. Then to conclude our proof, we establish the following main theorem: there is an equivalence between (i) and (ii): (i) (F0): a 1 4 + ( 2 s b 1 ) 4 = c 1 4 , is solvable in ℕ , with 2≤s∈ℕ , ( a 1 , b 1 , c 1 )∈ ( 2ℕ+1 ) 3 , coprime in pairs. (ii) ∃( a 1 , b 1 , c 1 )∈ ( 2ℕ+1 ) 3 , coprime in pairs, for wich: ∃( b ′ 2 , b 2 , b ″ 2 )∈ ( 2ℕ+1 ) 3 coprime in pairs, and 2≤s∈ℕ , checking b 1 = b ′ 2 b 2 b ″ 2 , and such that for notations: S=s−λ( s−1 ) , with λ∈{ 0,1 } defined by c 1 − a 1 2 ≡λ( mod2 ) , d b =gcd( 2 s b 1 , c 1 − a 1 )= 2 S b 2 and d ′ b = 2 s−S b ′ 2 = 2 s B 2 d b , where ( 2 s B 2 ) 2 =gcd( b 1 2 , c 1 2 − a 1 2 ) , the following system is checked: { c 1 − a 1 = d b 4 2 2+λ = 2 2−λ ( 2 S−1 b 2 ) 4 c 1 + a 1 = 2 1+λ d ′ b 4 = 2 1+λ ( 2 s−S b ′ 2 ) 4 c 1 2 + a 1 2 =2 b ″ 2 4;and this system implies: ( b 1−λ,2 4 ) 2 + ( 2 4s−3 b λ,2 4 ) 2 = ( b ″ 2 2 ) 2;where: ( b 1−λ,2 , b λ,2 , b ″ 2 )={ ( b ′ 2 , b 2 , b ″ 2 ) if λ=

关键词： Factorisation in ℤ Greatest Common Divisor Pythagoras Equation Pythagorician Triplets Fermat's Equations Pythagorician Divisors Fermat's Divisors Diophantine Equations of Degree 2 4-Integral Closure of ℤ in ℚ

Irrigated Agriculture Facing the Challenge of Climate Change: Adaptation Strategies for Farmers in the Irrigated Perimeters of Môle Saint-Nicolas, Haiti

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American Journal of Climate Change 2024年第3期13卷 477-498页

作者： frantz Mial Jean-françois Bissonnette Marc-André Bourgault Meryem Qacami Department of Geography Laval University Quebec Canada

Môle Saint-Nicolas, like all other communes in the Republic of Haiti, faces increasing climate variability, impacting agricultural production and water resources. Consequently, there is a pressing need for adaptation to these climatic changes. This research aims to showcase the adaptation strategies deployed by farmers to cope with the increasing climate variability. Surveys were conducted through group and individual discussions with a randomly selected cohort of 150 farmers. Two types of analysis were performed: quantitative and qualitative. The quantitative data analysis was conducted using Statistical Package for the Social Sciences (SPSS) software. The findings reveal that farmers have perceived changes in rainfall patterns, temperature, wind, and their environment. These changes manifest as irregular rainfall, higher temperatures, prolonged drought periods, violent winds accompanied by rain, premature cessation of rains, and reduced flow from water sources. In response, the most common adaptation strategies adopted include selecting new cultivars, early-maturing varieties, crop rotation and diversification, canal dredging, new soil preparation methods, upstream water source protection, and micro-watershed management. The significance of this research lies in its contribution to enhancing farmers’ adaptive capacities by alerting stakeholders in the irrigated perimeters about the consequences of climate change, thereby incorporating the real needs of farmers in future projects.

关键词： Climate Change Perception Adaptation Agriculture Irrigated Perimeter

Toward a Malaria-Free Future

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Beijing Review 2024年第23期67卷 30-31页

作者： francois Essomba ChinAfrica magazine

Malaria is a potentially fatal disease transmitted by certain species of *** to the World Health Organization(WHO),globally in 2022,there were an estimated 249 million malaria cases and 608,000 malaria deaths in 85 *** was the location of 94 percent of malaria cases and 95 percent of malaria deaths,with children under 5 there accounting for about 80 percent of all malaria deaths.

关键词： Malaria malaria cases

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A Taste of Success

ChinAfrica 2024年第5期16卷 44-45页

作者： franCois ESSOMBA Cameroon

Cooperation between China and Cameroon has reached extraordinary heights over the past two decades,which has deepened both the diplomatic and economic *** mutually beneficial partnership has made China the leading buyer of Cameroonian products,with a market share of 25.8 percent in 2021,and the leading supplier,with a market share of 17.5 percent in 2020,according to data from the National Institute of Statistics of Cameroon.

关键词： mutually market partnership

An ethically guided preclinical device for phenotyping H_(2)production in laboratory rodents

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Animal Models and Experimental Medicine 2024年第4期7卷 553-561页

作者： Victor Pascal-Moussellard Emilie Boucher Stéphane Tanguy Philippe Cinquin Pierre-Alain Barraud ChloéDavin Cordélia Salomez-Ihl Dalil Hannani françois Boucher Jean-Pierre Alcaraz University Grenoble Alpes CNRSUMR 5525VetAgro SupGrenoble INPTIMCGrenobleFrance

Background:Dihydrogen(H_(2))is produced endogenously by the intestinal microbiota through the fermentation of diet *** the past few years,numer-ous studies have demonstrated the significant therapeutic potential of H_(2)in various pathophysiological contexts,making the characterization of its production in labora-tory species of major preclinical ***:This study proposes an innovative solution to accurately monitor H_(2)pro-duction in free-moving rodents while respecting animal welfare *** devel-oped device consisted of a wire rodent cage placed inside an airtight chamber in which the air quality was maintained,and the H_(2)concentration was continuously *** the airtightness and efficiency of the systems used to control and maintain air quality in the chamber were checked,tests were carried out on rats and mice with different metabolic phenotypes,over 12 min to 1-h experiments and repeatedly.H_(2)production rates(HPR)were obtained using an easy calculation algorithm based on a first-order moving ***:HPR in hyperphagic Zucker rats was found to be twice as high as in control Wistar rats,respectively,2.64 and 1.27 nmol.s^(−1)per *** addition,the ingestion of inulin,a dietary fiber,stimulated H_(2)production in *** were 0.46 nmol.s^(−1)for animals under control diet and 1.99 nmol.s^(−1)for animals under inulin ***:The proposed device coupled with our algorithm enables fine analysis of the metabolic phenotype of laboratory rats or mice with regard to their endogenous H_(2)production.

关键词： hydrogen therapy laboratory rodents microbiota molecular hydrogen noninvasive monitoring device

Measurement of Radon Concentration and Estimation of Cancer Risk in Twenty-Four Model Houses in the Town of Koudougou

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Open Journal of Applied Sciences 2024年第1期14卷 193-204页

作者： Moumouni Derra Luc Telado Bambara Karim Kaboré Yalgado Zakaria Sawadogo Ousmane Cissé françois Zougmoré Physics Department University Norbert Zongo Koudougou Burkina Faso Physics and Chemical Department Institute of Sciences and Technology “Ecole Normale Supérieure” Ouagadougou Burkina Faso Physics Department Virtual University Ouagadougou Burkina Faso Laboratory of Materials and Environment University Joseph Ki-Zerbo Ouagadougou Burkina Faso

The objective of our study is to evaluate the concentration of radon (86Rn) inside houses in the town of Koudougou in order to estimate its impact on the health of the population. Indeed, when uranium-rich minerals are found near the surface of the ground, radon concentrations can reach tens of becquerels per cubic meter in enclosed spaces. Given the nature of the geological base of Burkina Faso, this situation is quite probable and certain places that are sometimes poorly ventilated (house, school, office, etc.) can have radon levels high enough to constitute a health problem for occupants. Thus, twenty-four (24) sample houses were identified. In each house, the Corentium digital detector was between 0.8 m and 2 m for at least one week in a place where the occupants estimate that they spend more time of time and measure the concentration of radon in the long term and short term. The recorded data allowed us to determine the Absorbed Dose and the Annual Effective Dose of radon gas for each house in order to estimate the Risk of Cancer and the probable Number of Cases of Lung Cancer per million inhabitants. Thus, the results indicate that the long-term radon concentration varies between 6 Bq/m3 and 285 Bq/m3 respectively in houses 11 and 4 compared to 1 Bq/m3 to 208 Bq/m3 in the short term in the same houses. Also, in the long term, in control houses 1, 3 and 4, the radon level is above the recommended threshold interval. For the short term, these are houses 1, 3, 4 and 17 respectively with 110 Bq/m3, 142 Bq/m3, 208 Bq/m3 and 105 Bq/m3. As for the long-term and short-term effective doses, only houses 1, 3, 4, 17 and 24 have values between 3 - 10 Sv/year. The estimation of the relative risk of lung cancer gives values relatively close to unity and between 1.006 and 1.142 with an average of 1.035 and that of the Number of Lung Cancer Cases per million inhabitants gives va

关键词： Radon Absorbed Dose Annual Effective Dose Cancer Risk

Estimating the Input Power of a Power Plant Using the Efficiency of the Inverter

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Smart Grid and Renewable Energy 2024年第3期15卷 99-106页

作者： Toussaint Tilado Guingane Sosthène Tassembedo Eric Korsaga Dominique Bonkoungou Zacharie Koalaga françois Zougmore Physics UFR/SEA Materials and Environment Laboratory (LAME) Pr. Joseph Ki ZERBO University Ouagadougou Burkina Faso Physics UFR/ST Science and Technology Laboratory (LaST) Thomas SANKARA University Ouagadougou Burkina Faso

The study focuses on estimating the input power of a power plant from available data, using the theoretical inverter efficiency as the key parameter. The paper addresses the problem of missing data in power generation systems and proposes an approach based on the efficiency formula widely documented in the literature. In the absence of input data, this method makes it possible to estimate the plant’s input power using data extracted from the site, in particular that provided by the Ministry of the Environment. The importance of this study lies in the need to accurately determine the input power in order to assess the overall performance of the energy system.

关键词： Estimation Data Missing Input Power Efficiency Inverter