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

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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On Fermat Last Theorem: The New Efficient Expression of a Hypothetical Solution as a Function of Its Fermat Divisors

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American Journal of Computational Mathematics 2023年第1期13卷 82-90页

作者： prosper kouadio kimou Département Mathématiques et Informatiques Institut Polytechnique Félix Houphou&#235t-Boigny Yamoussoukro C&#244te d’Ivoire

Denote by a non-trivial primitive solution of Fermat’s equation (p prime).We introduce, for the first time, what we call Fermat principal divisors of the triple defined as follows. , and . We show that it is possible to express a,b and c as function of the Fermat principal divisors. Denote by the set of possible non-trivial solutions of the Diophantine equation . And, let (p prime). We prove that, in the first case of Fermat’s theorem, one has . In the second case of Fermat’s theorem, we show that , ,. Furthermore, we have implemented a python program to calculate the Fermat divisors of Pythagoreans triples. The results of this program, confirm the model used. We now have an effective tool to directly process Diophantine equations and that of Fermat. .

关键词： Fermat’s Last Theorem Fermat Divisors Barlow’s Relations Greatest Common Divisor

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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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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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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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Biological Control of Erosion of Banana Drains in Côte D’ivoire

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Agricultural Sciences 2016年第7期7卷 493-501页

作者： kouadio Y. prosper Boraud N’Takpé Kama Maxime Tiébré Marie-Solange Djakalia Ouattara Edouard Kouakou N’guessan Laboratory of Botany UFR Biosciences University Félix Houphou&#235 t-Boigny Abidjan C&#244 te d’Ivoire

The erosion of drains is a major limitation of the quality, the increasing of banana production and the environmental protection of industrial banana in Côte d'Ivoire. It leads inundations, death of banana trees and significant loss of production. Thence, the construction and the maintenance of drain costs too much and causes injure, snake bite, physical traumatisms, many diseases, … These events compromise the sustainable production of banana by reducing seriously worker’s the activities and finally increase the cost of production. The aim of the present work is to contribute to the sustainable development and human capacity building in the third world nations as far as banana production is concerned. The methods used so far to address this phenomenon proved inefficient. The technology innovation in this area has been to grow grass on the outer edges of the channels drained water. This resulted in a systematic reduction of erosion. Better still, it helped fertilize the soil, reduce the deportations of fertilizer and improve the quality of landscape of the plantations. Stenotaphrum secondatum is the best vegetable specie adapted to the biological control against water erosion of drains.

关键词： Banana Biological Control Ivory Coast Drain Environment

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