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

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

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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

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An Infinite Family of Number Fields with No Inert Primes

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Advances in Pure Mathematics 2022年第12期12卷 744-756页

作者： françois emmanuel tanoé UFR Mathematics and Computer Science Félix Houphouet-BOIGNY University Abidjan Ivory Coast

The goal of this paper is to show that there are infinitely many number fields K/Q, for which there is no inert prime p ∈ N*, i.e. ∀p ∈ N* a prime number, prime ideal of K such that where: Zk is the Dedekind domain of the integer elements of K. To prove such a result, consider for any prime p, the decomposition into a product of prime ideals of Zk, of the ideal . From this point, we use on the one hand: 1) The well- known property that says: If , then the ideal pZk decomposes into a product of prime ideals of Zk as following: . (where:;is the irreducible polynomial of θ, and, is its reduction modulo p, which leads to a product of irreducible polynomials in Fp[X]). It is clear that because if is reducible in Fp[X], then consequently p is not inert. Now, we prove the existence of such p, by proving explicit such p as follows. So we use on the other hand: 2) this property that we prove, and which is: If , is an irreducible normalized integer polynomial, whose splitting field is , then for any prime number p ∈ N: is always a reducible polynomial. 3) Consequently, and this closes our proof: let’s consider the set (whose cardinality is infinite) of monogenic biquadratic number fields: . Then each fθ(X) checks the above properties, this means that for family M, all its fields, do not admit any inert prime numbers p ∈ N. 2020-Mathematics Subject Classification (MSC2020) 11A41 - 11A51 - 11D25 - 11R04 - 11R09 - 11R11 - 11R16 - 11R32 - 11T06 - 12E05 - 12F05 -12F10 -13A05-13A15 - 13B02 - 13B05 - 13B10 - 13B25 -13F05

关键词： Fields Extensions Splitting Fields Polynomials Finite Fields Extensions Polynomials of Fp[X] Dedekind Ring Ramification Theory Monogeneity Quadratic & Biquadratic Fields Irreducible Polynomials of Degree 3 & 4

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Diophantine Quotients and Remainders with Applications to Fermat and Pythagorean Equations

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American Journal of Computational Mathematics 2023年第1期13卷 199-210页

作者： Prosper Kouadio Kimou françois emmanuel tanoé Department of Mathematics and Computer Science Félix Houphouet-Boigny National Polytechnic Institute Yamoussoukro Ivory Coast UFR Mathematics of Computer Science Université Félix Houphouet-Boigny Abidjan Ivory Coast

Diophantine equations have always fascinated mathematicians about existence, finitude, and the calculation of possible solutions. Among these equations, one of them will be the object of our research. This is the Pythagoras’- Fermat’s equation defined as follows. (1) when , it is well known that this equation has an infinity of solutions but has none (non-trivial) when . We also know that the last result, named Fermat-Wiles theorem (or FLT) was obtained at great expense and its understanding remains out of reach even for a good fringe of professional mathematicians. The aim of this research is to set up new simple but effective tools in the treatment of Diophantine equations and that of Pythagoras-Fermat. The tools put forward in this research are the properties of the quotients and the Diophantine remainders which we define as follows. Let a non-trivial triplet () solution of Equation (1) such that . and are called the Diophantine quotients and remainders of solution . We compute the remainder and the quotient of b and c by a using the division algorithm. Hence, we have: and et with . We prove the following important results. if and only if and if and only if . Also, we deduce that or for any hypothetical solution . We illustrate these results by effectively computing the Diophantine quotients and remainders in the case of Pythagorean triplets using a Python program. In the end, we apply the previous properties to directly prove a partial result of FLT. .

关键词： Diophantine Equation Modular Arithmetic Fermat-Wiles Theorem Pythagorean Triplets Division Theorem Division Algorithm Python Program Diophantine Quotients Diophantine Remainders

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Pythagorician Divisors and Applications to Some Diophantine Equations

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Advances in Pure Mathematics 2023年第2期13卷 35-70页

作者： françois emmanuel tanoé Prosper Kouadio Kimou UFR Mathématiques et Informatique. Université Félix Houphouë t BOIGNY Abidjan C&#244te D’ivoire Laboratoire d’Informatique et de Mathématiques Appliquées. Institut Polytechnique Félix Houphouë t BOIGNY Yamoussoukro C&#244te D’ivoire

We consider the Pythagoras equation X2 +Y2 = Z2, and for any solution of the type (a,b = 2sb1 ≠0,c) ∈ N*3, s ≥ 2, b1odd, (a,b,c) ≡ (±1,0,1)(mod 4), c > a , c > b, and gcd(a,b,c) = 1, we then prove the Pythagorician divisors Theorem, which results in the following: , where (d,d′′) (resp. (e,en)) are unique particular divisors of a and b, such that a = dd′′ (resp. b = ee′′ ), these divisors are called: Pythagorician divisors from a, (resp. from b). Let’s put λ ∈{0,1}, defined by: and S = s -λ (s -1). Then such that . Moreover the map is a bijection. We apply this new tool to obtain a new classification of the primitive, positive and non-trivial solutions of the Pythagoras equations: a2 + b2 = c2 via the Pythagorician parameters (d,e,S ). We obtain for (d, e) fixed, the equivalence class of any Pythagorician solution (a,b,c), checking , namely: . We also update the solutions of some Diophantine equations of degree 2, already known, but very important for the resolution of other equations. With this tool of Pythagorean divisors, we have obtained (in another paper) new recurrent methods to solve Fermat’s equation: a4 + b4 = c4, other than usual infinite descent method;and to solve congruent numbers problem. We believe that this tool can bring new arguments, for Diophantine resolution, of the general equations of Fermat: a2p + b2p = c2p and ap + bp = cp. MSC2020-Mathematical Sciences Classification System: 11A05-11A51-11D25-11D41-11D72.

关键词： Pythagoras Equation Pythagorician Triplets Diophantine Equations of Degree 2 Factorisation-Gcd-Fermat’s Equations

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Experimental and Numerical Analysis of a PCM-Integrated Roof for Higher Thermal Performance of Buildings

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Journal of Thermal Science 2024年第2期33卷 522-536页

作者： SIMON françois RUIZ-VALERO Letzai GIRARD Aymeric GALLEGUILLOS Hector Centro de Desarrollo Urbano Sustentable(CEDEUS) Pontificia Universidad Católica de Chile7500000SantiagoChile Facultad de Ciencias e Ingeniería Pontificia Universidad Católica Madre y Maestra(PUCMM)10103Santo DomingoDominican Republic LMGC IMT Mines AlesUniv MontpellierCNRS30100AlèsFrance Facultad de Ingeniería y Ciencias Universidad Adolfo Ibañez2520000Viña del MarChile Centro de Desarrollo Energético Antofagasta(CDEA) Universidad de Antofagasta1240000AntofagastaChile

Phase change materials(PCMs) designate materials able to store latent *** change state from solid to liquid over a defined temperature *** process is reversible and can be used for thermo-technical *** present paper aims to study the thermal performance of an inorganic eutectic PCM integrated into the rooftop slab of a test room and analyze its potential for building thermal *** experiment is conducted in two test rooms in Antofagasta(Chile) during summer,fall,and *** PCM is integrated into the rooftop of the first test room,while the roof panel of the second room is a sealed air *** work introduces a numerical model,which is built using the finite difference method and used to simulate the rooms' thermal *** thermal simulations of the PCM room are performed for other Chilean locations to evaluate and compare the capability of the PCM panel to store latent heat thermal energy in different *** show that the indoor temperature of the PCM room in Antofagasta varies only 21.1℃±10.6℃,while the one of the air-panel room varies 28.3℃±18.5℃.Under the experiment's conditions,the PCM room's indoor temperature observes smoother diurnal fluctuations,with lower maximum and higher minimum indoor temperatures than that of the air-panel *** simulations in other cities show that the PCM panel has a better thermal performance during winter,as it helps to maintain or increase the room temperature by some degrees to reach comfort *** demonstrates that the implementation of such PCM in the building envelope can effectively reduce space heating and cooling needs,and improve indoor thermal comfort in different climates of Chile.

关键词： phase change material thermal energy storage building space heating and cooling energy efficiency designbuilder

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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

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Improving the utility of locally differentially private protocols for longitudinal and multidimensional frequency estimates

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Digital Communications and Networks 2024年第2期10卷 369-379页

作者： Héber H.Arcolezi Jean-françois Couchot Bechara Al Bouna Xiaokui Xiao Inria andÉcole Polytechnique(IPP) PalaiseauFrance Femto-ST Institute Univ.Bourg.Franche-ComtéUBFCCNRSBelfortFrance TICKET Lab. Antonine University Hadat-BaabdaBaabdaLebanon School of Computing National University of SingaporeSingaporeSingapore

This paper investigates the problem of collecting multidimensional data throughout time(i.e.,longitudinal studies)for the fundamental task of frequency estimation under Local Differential Privacy(LDP)*** to frequency estimation of a single attribute,the multidimensional aspect demands particular attention to the privacy ***,when collecting user statistics longitudinally,privacy progressively ***,the“multiple”settings in combination(i.e.,many attributes and several collections throughout time)impose several challenges,for which this paper proposes the first solution for frequency estimates under *** tackle these issues,we extend the analysis of three state-of-the-art LDP protocols(Generalized Randomized Response–GRR,Optimized Unary Encoding–OUE,and Symmetric Unary Encoding–SUE)for both longitudinal and multidimensional data *** the known literature uses OUE and SUE for two rounds of sanitization(***),i.e.,L-OUE and L-SUE,respectively,we analytically and experimentally show that starting with OUE and then with SUE provides higher data utility(i.e.,L-OSUE).Also,for attributes with small domain sizes,we propose Longitudinal GRR(L-GRR),which provides higher utility than the other protocols based on unary ***,we also propose a new solution named Adaptive LDP for LOngitudinal and Multidimensional FREquency Estimates(ALLOMFREE),which randomly samples a single attribute to be sent with the whole privacy budget and adaptively selects the optimal protocol,i.e.,either L-GRR or *** shown in the results,ALLOMFREE consistently and considerably outperforms the state-of-the-art L-SUE and L-OUE protocols in the quality of the frequency estimates.

关键词： Local differential privacy Discrete distribution estimation Frequency estimation Multidimensional data Longitudinal studies

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A mediation system for continuous spatial queries on a unified schema using Apache Spark

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Big Earth Data 2024年第1期8卷 115-141页

作者： Thi Thu Trang Ngo françois Pinet David Sarramia Myoung-Ah Kang UniversitéClermont Auvergne ISIMAAubièreFrance UniversitéClermont Auvergne INRAEUR TSCFClermont-FerrandFrance UniversitéClermont Auvergne CNRS/IN2P3LPCClermont-FerrandFrance

Recent advances in big and streaming data systems have enabled real-time analysis of data generated by Internet of Things(IoT)systems and sensors in various *** this context,many applications require integrating data from several heterogeneous sources,either stream or static *** such as Apache Spark are able to integrate and process large datasets from different ***,these frameworks are hard to use when the data sources are heterogeneous and *** address this issue,we propose a system based on mediation techniques for integrating stream and static data *** integration process of our system consists of three main steps:configuration,query expression and query *** the configuration step,an administrator designs a mediated schema and defines mapping between the mediated schema and local data *** the query expression step,users express queries using customized SQL grammar on the mediated ***,our system rewrites the query into an optimized Spark application and submits the application to a Spark *** results are continuously returned to *** experiments show that our optimizations can improve query execution time by up to one order of magnitude,making complex streaming and spatial data analysis more accessible.

关键词： Streaming data streaming data integration mediator geospatial data continuous queries

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Template-free nanostructured particle growth via a one-pot continuous gradient nanoprecipitation

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Aggregate 2024年第1期5卷 409-418页

作者： Xie Cheng Shuai Wang Julien Bernard françois Ganachaud Xibo Yan School of Chemical Engineering and Technology Tianjin UniversityTianjinChina College of Food Science and Engineering Tianjin University of Science and TechnologyTianjinChina Universitéde Lyon CNRSUniversitéClaude Bernard Lyon 1INSA-LyonUniversitéJean MonnetUMR5223Ingénierie des Matériaux PolymèresVilleurbanne CedexFrance

Engineered nanoparticles have emerged as new types of materials for a wide range of applications from therapeutics to ***,fabricating nanomaterials presenting complex inner morphologies and shapes in a simple manner remains a great ***,we report the template-free one-pot continuous gradient nanoprecipitation of different types of non-compatible polymers to spontaneously form nanostructured *** continuous addition of antisolvent induces precipitation and(re)organization of polymer chains at the forming particle interface,ultimately and naturally developing complex inner morphologies and shapes while particle *** low-energy-cost bottom-up assembly approach applies to various functional polymers,possibly embedded with metal nanoparticles,for continuous growth into well-organized *** crosslinking of the particles and core removal allows both confirming the building process and leading to hollow or multivoid nanomaterials.

关键词： aggregation nanoprecipitation phase diagrams polymeric nanoparticles self-assembly

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