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The Fermat Classes and the Proof of Beal Conjecture Mohamed Sghiar

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The Fermat Classes and the Proof of Beal Conjecture Mohamed Sghiar The Fermat classes and the proof of Beal conjecture Mohamed Sghiar To cite this version: Mohamed Sghiar. The Fermat classes and the proof of Beal conjecture. IOSR Journal of Mathematics(IOSR-JM), International Organization Of Scientific Research, In press. hal-02878292 HAL Id: hal-02878292 https://hal.archives-ouvertes.fr/hal-02878292 Submitted on 23 Jun 2020 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. M. Sghiar, Mohamed Sghiar. "The Fermat Classes And The Proof Of Beal Conjecture." IOSR Journal of Mathematics (IOSR-JM), 16(3), (2020): pp. 09-13 The Fermat classes and the proof of Beal conjecture Par : Mohamed Sghiar [email protected] Présenté à : UNIVERSITÉ DE BOURGOGNE DIJON Faculté des sciences Mirande Département de mathématiques et informatiques 9 Av Alain Savary 21078 DIJON CEDEX Abstract : If after 374 years the famous theorem of Fermat-Wiles was demonstrated in 150 pages by A. Wiles [4], The purpose of this article is to give a proofs both for the Fermat last theorem and the Beal conjecture by using the Fermat class concept. Résumé : Si après 374 ans le célèbre théorème de Fermat-Wiles a été dé- montré en 150 pages par A. Wiles [4], le but de cet article est de donner des démonstrations à la fois du dernier théorème de Fermat et de la conjecture de beal en utilisant la notion des classes de Fermat. Keywords : Fermat, Fermat-Wiles theorem, Fermat’s great theorem, Beal conjecture, Diophantine equation The Subject Classification Codes : 11D41 - 11G05 - 11G07 - 26B15 - 26B20 - 28A10 - 28A75 - 26A09 - 26A42 - 1 M. Sghiar, Mohamed Sghiar. "The Fermat Classes And The Proof Of Beal Conjecture." IOSR Journal of Mathematics (IOSR-JM), 16(3), (2020): pp. 09-13 1 Introduction, notations and definitions Set out by Pierre de Fermat [3], it was not until more than three centuries ago that Fermat’s great theorem was published, validated and established by the British mathematician Andrew Wiles [4] in 1995. In mathematics, and more precisely in number theory, the last theorem of Fermat [3], or Fermat’s great theorem, or since his Fermat-Wiles theorem demonstration [4], is as follows: There are no non-zero integers a, b, and c such that: an + bn = cn, as soon as n is an integer strictly greater than 2 ". The Beal conjecture [2] is the following conjecture in number theory: If ax + by = cz where a, b, c, x, y and z are positive integers with x, y, z > 2, then a, b, and c have a common prime factor. Equivalently, There are no solutions to the above equation in positive integers a, b, c, x, y, z with a, b and c being pairwise coprime and all of x, y, z being greater than 2. The purpose of this article is to give a proofs both for the Fermat last theorem and the Beal conjecture by using the Fermat class concept. Let be two equations xa + yb − zc = 0 with (x, y, z) ∈ E3 and (a, b, c) ∈ F 3, and XA + Y B − ZC = 0 with (X, Y, Z) ∈ E03 and (A, B, C) ∈ F 03,in the following F = F 0 = N and E and E’ are subsets of R . The two equations xa + yb − zc = 0 with (x, y, z) ∈ E3 and (a, b, c) ∈ F 3; and XA + Y B − ZC = 0 with (X, Y, Z) ∈ E03 and (A, B, C) ∈ F 03, are said to be equivalent if the resolution of one is reduced to the resolution of the other. In the following, an equation xa + yb − zc = 0 with (x, y, z) ∈ E3 and (a, b, c) ∈ F 3 is considered at close equivalence, we say xa + yb − zc = 0 is a Fermat class. Example: The equation x15 + y15 − z15 = 0 with (x, y, z) ∈ Q3 is equivalent 3 3 3 3 to the equation X + Y − Z = 0 with (X, Y, Z) ∈ Q5 and where Q5 = {q5, q ∈ Q} 2 M. Sghiar, Mohamed Sghiar. "The Fermat Classes And The Proof Of Beal Conjecture." IOSR Journal of Mathematics (IOSR-JM), 16(3), (2020): pp. 09-13 2 The proof of Fermat’s last theorem Theorem 1. There are no non-zero a, b, and c three elements of E with E ⊂ Q such that: an + bn = cn, with n an integer strictly greater than 2 Lemma 1. If n ∈ N, a, b and c are a non-zero three elements of R with an + bn = cn then: Z b c − a n−1 c − a xn−1 − x + a dx = 0 0 b b Proof. Z a Z b Z c an + bn = cn ⇐⇒ nxn−1dx + nxn−1dx = nxn−1dx 0 0 0 But as : Z c Z a Z c nxn−1dx = nxn−1dx + nxn−1dx 0 0 a So : Z b Z c nxn−1dx = nxn−1dx 0 a And as by changing variables we have : Z c Z b c − a n−1 c − a nxn−1dx = n y + a dy a 0 b b Then : Z b Z b c − a n−1 c − a xn−1dx = y + a dy 0 0 b b It results: Z b c − a n−1 c − a xn−1 − x + a dx = 0 0 b b 3 M. Sghiar, Mohamed Sghiar. "The Fermat Classes And The Proof Of Beal Conjecture." IOSR Journal of Mathematics (IOSR-JM), 16(3), (2020): pp. 09-13 Corollary 1. If N, n ∈ N∗, a, b and c are a non-zero three elements of R and an + bn = cn then : b n−1 Z N c − a a c − a xn−1 − x + dx = 0 0 b N b a b Proof. It results from lemma 1 by replacing a, b and c respectively by N , N c and N Lemma 2. If an + bn = cn is a Fermat class, where n ∈ N, a, b and c are a non-zero three elements of E ⊂ R+ with n > 2 and 0 < a ≤ b ≤ c. Then we can choose a not zero integer N, a, b , c and n in the class, such that : c − a a n−1 c − a " b # f(x) = xn−1 − x + ≤ 0 ∀x ∈ 0, b N b N Proof. df c − a a n−2 c − a2 = (n − 1)xn−2 − (n − 1) x + dx b N b The function f decreases to the right of 0 in [0, [. 1 a c−a n−1 N ( b ) But f(x) = 0 ⇐⇒ x = 1 c−a 1+ n−1 1−( b ) 1 a c−a n−1 N ( b ) So f(x) ≤ 0 ∀x such that 0 ≤ x ≤ 1 c−a 1+ n−1 1−( b ) 1 a c−a n−1 N ( b ) And f(x) ≥ 0 ∀x such that x ≥ 1 c−a 1+ n−1 1−( b ) 1 b(1−µ) a ( c−a ) n−1 1 N b c−a 1+ n−1 Otherwise if µ ∈]0, 1] we have 1 ⇐⇒ 1−µ(1−( ) ) N c−a 1+ n−1 b 1−( b ) 1 1 c−a 1+ n−1 a c−a n−1 ( b ) + b ( b ) 0 1 0 1 1 By replacing a, b and c respectively with a = a k , b = b k , and c k , we get another Fermat class : a0kn + b0kn = c0kn 4 M. Sghiar, Mohamed Sghiar. "The Fermat Classes And The Proof Of Beal Conjecture." IOSR Journal of Mathematics (IOSR-JM), 16(3), (2020): pp. 09-13 0 0 1 c −a 1+ kn−1 we will show for this class and for k large enough that 1−µ(1−( b0 ) ) ≤ 0 0 1 0 0 0 1 c −a 1+ kn−1 a c −a kn−1 ( b0 ) + b0 ( b0 ) : 0 0 1 0 0 c −a 1+ kn−1 c −a 2 First we have : 1 − µ(1 − ( b0 ) ) ≤ 1 − µ(1 − ( b0 ) ) 1 1 1 1 1 kn−1 1 0 0 1 0 0 0 1 0 0 0 1 k k kn−1 c −a 1+ kn−1 a c −a kn−1 c c −a kn−1 c −a a k And as ( b0 ) + b0 ( b0 ) = b0 ( b0 ) ≥ ( 1 ) ≥ (1 − ( b ) ) b k 1 1 1 kn−1 1 kn By using the logarithm, we have lim (1 − ( a ) k ) = lim (1 − ( a ) k ) = k→+∞ b k→+∞ b 1 because : 1 1 1 1 a k 1 a k a k k ln(1−( b ) ) k −N (1 − ( b ) ) = e , by posing : 1 − ( b ) = e , we will have : 1 ln(1−e−N ) −N 1 k −N a 1 ln(1−e ) a k ln( ) = a and lim (1 − ( ) ) = lim e b = 1 which shows k ln( b ) k→+∞ b N →+∞ the result. So, for k large enough, we deduce that there exists a class a0kn + b0kn = c0kn such that : c0 − a0 a0 !kn−1 c0 − a0 " b0(1 − µ)# f(x) = xkn−1 − x + 0 ∀x ∈ 0, b0 N b0 N independently of N. h b0(1−µ) i Let’s fix an N and put S = sup{f(x), x ∈ 0, N } By replacing a0, b0 and c0 respectively with a0(1 − µ) = a00, b0(1 − µ) = b00, and c0(1 − µ) = c00, we get another Fermat class : a00kn + b00kn = c00kn And we will have for M large enough : c00 − a00 a00 !kn−1 c00 − a00 " b00 # f(x) = xkn−1 − x + 0 ∀x ∈ 0, b00 M b00 M P (x,µ) h b0(1−µ) i Because f(x) ≤ S + Sup{ M , x ∈ 0, N } where P is a polynomial, P (x,µ) h b0(1−µ) i and as for M large enough |Sup{ M , x ∈ 0, N }| |S| and S 0, the result is deduced.
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