Pythagorician Divisors and Applications to Some Diophantine Equations  

作　　者：François Emmanuel Tanoé Prosper Kouadio Kimou François Emmanuel Tanoé;Prosper Kouadio Kimou(UFR Mathématiques et Informatique. Université Félix Houphouë,t BOIGNY, Abidjan, Cô,te D’ivoire;Laboratoire d’Informatique et de Mathématiques Appliquées. Institut Polytechnique Félix Houphouë,t BOIGNY, Yamoussoukro, Cô,te D’ivoire)

机构地区：[1]UFR Mathématiques et Informatique. Université Félix Houphouë,t BOIGNY, Abidjan, Cô,te D’ivoire [2]Laboratoire d’Informatique et de Mathématiques Appliquées. Institut Polytechnique Félix Houphouë,t BOIGNY, Yamoussoukro, Cô,te D’ivoire

出　　处：《Advances in Pure Mathematics》2023年第2期35-70,共36页理论数学进展（英文）

摘　　要：We consider the Pythagoras equation X² +Y² = Z², and for any solution of the type (a,b = 2^sb₁≠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: a^2p + b^2p = c^2p and a^p + b^p = c^p. MSC2020-Mathematical Sciences Classification System: 11A05-11A51-11D25-11D41-11D72.We consider the Pythagoras equation X² +Y² = Z², and for any solution of the type (a,b = 2^sb₁≠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: a^2p + b^2p = c^2p and a^p + b^p = c^p. 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 

分 类 号：O15[理学—数学]