DOCSLIB.ORG

Proof of Three Conjectures: Beal's Conjecture, Riemann

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Proof of Three Conjectures: Beal's Conjecture, Riemann PROOF OF THREE CONJECTURES: BEAL’S CONJECTURE, RIEMANN HYPOTHESIS AND ABC VERSION 1.0 - JUNE 2021 Abdelmajid Ben Hadj Salem, Ingénieur Général Abstract.— This monograph presents the proofs of 3 important conjectures in the field of Number theory: - The Beal’s conjecture. - The Riemann Hypothesis. - The abc conjecture. We give in detail all the proofs. Résumé.— Cette monographie présente les preuves de 3 conjectures im- portantes dans le domaine de la théorie des nombres: - La conjecture de Beal. - L’Hypothèse de Riemann. - La conjecture abc. Nous donnons les détails des différentes démonstrations. Abdelmajid Ben Hadj Salem, Ingénieur Général PROOFS OF THREE CONJECTURES IN NUMBER THEORY : BEAL’S CONJECTURE, RIEMANN HYPOTHESIS AND THE ABC CONJECTURE VERSION 1.0 - JUNE 2021 Abdelmajid Ben Hadj Salem, Ingénieur Général Résidence Bousten 8, Mosquée Raoudha, 1181 Soukra Raoudha, Tunisia, . E-mail : [email protected], c -2021- Abdelmajid Ben Hadj Salem - Figure 1. Photo of the Author (2011) To the memory of my Parents, to my wife Wahida, my daughter Sinda and my son Mohamed Mazen PROOFS OF THREE CONJECTURES IN NUMBER THEORY : BEAL’S CONJECTURE, RIEMANN HYPOTHESIS AND THE ABC CONJECTURE VERSION 1.0 - JUNE 2021 Abdelmajid Ben Hadj Salem, Ingénieur Général Abstract.— This monograph presents the proofs of 3 important conjectures in the field of Number theory: - The Beal’s conjecture. - The Riemann Hypothesis. - The abc conjecture. We give in detail all the proofs. Résumé.— Cette monographie présente les preuves de 3 conjectures importantes dans le domaine de la théorie des nombres à savoir: - La conjecture de Beal. - L’Hypothèse de Riemann. - La conjecture abc. Nous donnons les détails des différentes démonstrations. CONTENTS 1. A Complete Proof of Beal’s Conjecture......................................... 8 1.1. Introduction. 8 1.1.1. Trivial Case. 9 1.2. Preliminaries. 9 1.2.1. Expressions of the roots. 12 1.3. Preamble of the Proof of the Main Theorem. 15 θ 1 1.3.1. Case cos2 = ............................................................. 15 3 b 1.3.1.1. b = 1.................................................................... 15 1.3.1.2. b = 2.................................................................... 16 1.3.1.3. b = 3.................................................................... 16 θ a 1.3.2. Case a > 1, cos2 = ...................................................... 16 3 b 1.4. Hypothesis : {3|a and b|4p}................................................... 17 1.4.1. Case b = 2 and 3|a :......................................................... 17 1.4.2. Case b = 4 and 3|a :......................................................... 17 1.4.3. Case b = p and 3|a :......................................................... 17 1.4.4. Case b|p ⇒ p = b.p0, p0 > 1, b 6= 2, b 6= 4 and 3|a :.......................... 21 1.4.5. Case b = 2p and 3|a :........................................................ 25 1.4.6. Case b = 4p and 3|a :........................................................ 26 1.4.7. Case 3|a and b = 2p0 b 6= 2 with p0|p :....................................... 31 1.4.8. Case 3|a and b = 4p0 b 6= 2 with p0|p :....................................... 34 1.4.9. Case 3|a and b|4p :........................................................... 38 1.5. Hypothèse: {3|p and b|4p} .................................................... 42 1.5.1. Case b = 2 and 3|p :......................................................... 42 1.5.2. Case b = 4 and 3|p :......................................................... 42 1.5.3. Case: b 6= 2, b 6= 4, b 6= 3, b|p and 3|p :....................................... 42 1.5.4. Case b = 3 and 3|p :......................................................... 47 1.5.5. Case 3|p and b = p:.......................................................... 48 1.5.6. Case 3|p and b = 4p :........................................................ 48 1.5.7. Case 3|p and b = 2p :........................................................ 48 CONTENTS 6 1.5.8. Case 3|p and b 6= 3 a divisor of p :........................................... 48 1.5.9. Case 3|p and b|4p:........................................................... 58 1.6. Numerical Examples. 73 1.6.1. Example 1:. 73 1.6.2. Example 2:. 74 1.6.3. Example 3:. 74 1.7. Conclusion. 75 Bibliography............................................................................ 76 2. Towards A Solution of The Riemann Hypothesis............................. 77 2.1. Introduction. 77 2.1.1. The function ζ ............................................................... 78 2.1.2. A Equivalent statement to the Riemann Hypothesis. 79 1 2.2. Proof that the zeros of η(s) are on the critical line <(s) = ................... 80 2 1 2.2.1. Case σ = =⇒ 2σ = 1...................................................... 82 2 1 2.2.2. Case 0 < <(s) < ........................................................... 83 2 1 2.2.2.1. Case there is no zeros of η(s) with s = σ + it and 0 < σ < . 84 2 1 2.2.2.2. Case where there are zeros of η(s) with s = σ + it and 0 < σ < . 84 2 1 2.2.3. Case < <(s) < 1........................................................... 84 2 2.3. Conclusion. 85 Bibliography............................................................................ 87 3. Is The abc Conjecture True?...................................................... 88 3.1. Introduction and notations. 88 3.2. A Proof of the conjecture c < rad1.63(abc), case c = a + b....................... 89 0.63 3.2.1. µa ≤ rad (a) ............................................................. 90 0.63 3.2.2. µc ≤ rad (c) .............................................................. 90 0.63 0.63 3.2.3. µa > rad (a) and µc > rad (c)......................................... 90 0.63 1.63 0.63 1.63 3.2.3.1. Case: rad (c) < µc ≤ rad (c) and rad (a) < µa ≤ rad (a):. 90 1.63 1.63 3.2.3.2. Case: µc > rad (c) or µa > rad (a) ............................. 90 1.63 1.63 3.2.3.3. Case µc > rad (c) and µa > rad (a).............................. 97 3.3. The Proof of the abc conjecture................................................... 99 3.3.1. Case : ≥ 0.63 .............................................................. 99 3.3.2. Case: < 0.63............................................................... 99 3.3.2.1. Case: c > R............................................................. 99 3.3.2.2. Case: c < R.............................................................101 3.4. Conclusion........................................................................101 Bibliography............................................................................102 CONTENTS 7 List of figures...........................................................................104 List of Tables...........................................................................105 CHAPTER 1 A COMPLETE PROOF OF BEAL’S CONJECTURE Abstract.— In 1997, Andrew Beal announced the following conjecture: Let A, B, C, m, n, and l be positive integers with m, n, l > 2. If Am + Bn = Cl then A, B, and C have a common factor. We begin to construct the polynomial P (x) = (x − Am)(x − Bn)(x + Cl) = x3 − px + q with p, q integers depending of Am,Bn and Cl. 3 We resolve x − px + q = 0 and we obtain the three roots x1, x2, x3 as functions of p, q and a parameter θ. Since Am,Bn, −Cl are the only roots of x3 − px + q = 0, we discuss the conditions that x1, x2, x3 are integers and have or not a common factor. Three numerical examples are given. Résumé.— En 1997, Andrew Beal avait annoncé la conjecture suivante: Soient A, B, C, m, n, et l des entiers positifs avec m, n, l > 2. Si Am + Bn = Cl alors A, B, et C ont un facteur commun. Nous commençons par construire le polynôme P (x) = (x−Am)(x−Bn)(x+Cl) = x3 −px+q avec p, q des entiers qui dépendent de Am,Bn et Cl. Nous résolvons x3 − px + q = 0 et nous obtenons les trois racines x1, x2, x3 comme fonctions de p, q et d’un paramètre θ. Comme Am,Bn, −Cl sont les seules racines de x3 −px+q = 0, nous discutons les conditions pourque x1, x2, x3 soient des entiers. Trois exemples numériques sont présentés. 1.1. Introduction In 1997, Andrew Beal [4] announced the following conjecture : 1.2. PRELIMINARIES 9 Conjecture 1.1.— Let A, B, C, m, n, and l be positive integers with m, n, l > 2. If: (1.1) Am + Bn = Cl then A, B, and C have a common factor. In this paper, we give a complete proof of the Beal Conjecture. Our idea is to construct a polynomial P (x) of three order having as roots Am,Bn and −Cl with the condition (1.1). The paper is organized as follows. In Section 1, we begin with the trivial case where Am = Bn. In Section 2, we consider the polynomial P (x) = (x − Am)(x − Bn)(x + Cl) = x3 − px + q. We express the three roots of P (x) = x3 − px + q = 0 in function of two parameters ρ, θ that depend of Am,Bn,Cl. The Sections 3,4 and 5 are the main parts of the 4p θ paper. We write that A2m = cos2 . As A2m is an integer, it follows that 3 3 θ a cos2 must be written as where a, b are two positive coprime integers. We 3 b discuss the conditions of divisibility of p, a, b so that the expression of A2m is an integer. Depending of each individual case, we obtain that A, B, C have or not a common factor. We present three numerical examples in section 6 and we give conclusions in the last section. 1.1.1. Trivial Case. — We consider the trivial case when Am = Bn. The equation (1.1) becomes: (1.2) 2Am = Cl l ∗ m l l m then 2|C =⇒ 2|C =⇒ ∃ c ∈ N /C = 2c, it follows 2A = 2 c =⇒ A = 2l−1cl. As l > 2, then 2|Am =⇒ 2|A =⇒ 2|Bn =⇒ 2|B. The conjecture (3.1) is verified. We suppose in the following that Am > Bn. 1.2. Preliminaries ∗ ∗ Let m, n, l ∈ N > 2 and A, B, C ∈ N such: (1.3) Am + Bn = Cl 1.2. PRELIMINARIES 10 We call: P (x) = (x − Am)(x − Bn)(x + Cl) = x3 − x2(Am + Bn − Cl) (1.4) +x[AmBn − Cl(Am + Bn)] + ClAmBn Using the equation (1.3), P (x) can be written as: (1.5) P (x) = x3 + x[AmBn − (Am + Bn)2] + AmBn(Am + Bn) We introduce the notations: (1.6) p = (Am + Bn)2 − AmBn (1.7) q = AmBn(Am + Bn) As Am 6= Bn, we have : (1.8) p > (Am − Bn)2 > 0 Equation (1.5) becomes: (1.9) P (x) = x3 − px + q Using the equation (1.4), P (x) = 0 has three different real roots : Am,Bn and −Cl.
Load more

Recommended publications
  • Quantum Chemical Computational Methods Have Proved to Be An
    33005 R.A.Muneshwar and Shaikh Irfan/ Elixir Appl. Math. 83 (2015) 33005-33008 Available online at www.elixirpublishers.com (Elixir International Journal) Applied Mathematics Elixir Appl. Math. 83 (2015) 33005-33008 Proof of Beal’s Conjecture R.A.Muneshwar and Shaikh Irfan Department of Mathematics, NES Science College, SRTMU Nanded, India – 431602. ARTICLE INFO ABSTRACT Article history: This paper is devoted to obtain a proof of Beal’s conjecture. In this paper we have given Received: 28 April 2015; proof of Beal’s conjecture for the following two cases. Received in revised form: Case 1:- If (x, y, z) = (mn, m, mn+1), then same prime divides A, B and C 26 May 2015; ; . Accepted: 4 June 2015; Case 2:- If (x, y, z) = (m, m, m+1), then same prime divides A, B and C,; . Keywords © 2015 Elixir All rights reserved. Beal's Conjecture, Fermat last theorem, Natural number, Prime number. Introduction x y z Beal's conjecture is a conjecture in number theory: If A, B, C, x, y, and z are positive integers with x, y, z > 2, and A + B = C then A, B, and C have a common prime factor . In 1993 Andrew Beal discovered this conjecture while investigating generalizations of Fermat’s last theorem. Later Robert Tijdeman and Don Zagier invented the same conjecture. While "Beal conjecture" is the more commonly accepted reference, it has also been referred to as the "Tijdeman–Zagier conjecture" in one published article. Beal initially offered a prize of US $5,000 in 1997, gradually raising this amount up to US $1,000,000.
    [Show full text]
  • The Proof of the Beal's Conjecture
    The Bulletin of Society for Mathematical Services and Standards Online: 2014-12-01 ISSN: 2277-8020, Vol. 12, pp 21-28 doi:10.18052/www.scipress.com/BSMaSS.12.21 CC BY 4.0. Published by SciPress Ltd, Switzerland, 2014 The Proof of The Beal’s Conjecture Leszek W. Guła Lublin-POLAND Keywords Common Prime Factor, Diophantine Equations, Fermat Equation, Greatest Common Divisor, Newton Binomial Formula. Abstract 1. The truly marvellous proof of The Fermat's Last Theorem (FLT). 2. Two false proofs of FLT for 3. The incomplete proof of FLT for odd prime numbers 4. The proof of The Beal's Conjecture. 5. The Beal's Theorem. Dedicatory Dedicated to my Parents and my Brother I. INTRODUCTION The cover of this issue of the Bulletin is the frontispiece to a volume of Samuel de Fermat's 1670 edition of Bachet's Latin translation of Diophantus’s Arithmetica. This edition includes the marginalia of the editor's father, Pierre de Fermat. Among these notes one finds the elder Fermat's extraordinary comment in connection with the Pythagorean equation the marginal comment that hints at the existence of a proof (a demonstratio sane mirabilis) of what has come to be known as Fermat's Last Theorem. Diophantus's work had fired the imagination of the Italian Renaissance mathematician Rafael Bombelli, as it inspired Fermat a century later. Problem II.8 of the Diophantus's Arithmetica asks how a given square number is split into two other squares. Diophantus's shows how to solve this sum-of-squares problem for and , inasmuch as for all Thus for all relatively prime natural numbers such that We have a primitive Pythagorean triple the primitive triple because the numbers are coprime.
    [Show full text]
  • The Complexity of Number Theory Frank Vega Joysonic, Uzun Mirkova 5, Belgrade, 11000, Serbia [email protected]
    Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 25 February 2020 doi:10.20944/preprints202002.0379.v1 The Complexity of Number Theory Frank Vega Joysonic, Uzun Mirkova 5, Belgrade, 11000, Serbia [email protected] Abstract The Goldbach’s conjecture has been described as the most difficult problem in the history of Mathematics. This conjecture states that every even integer greater than 2 can be written as the sum of two primes. This is known as the strong Goldbach’s conjecture. The conjecture that all odd numbers greater than 7 are the sum of three odd primes is known today as the weak Goldbach conjecture. A major complexity class is NSPACE(S(n)) for some S(n). We show if the weak Goldbach’s conjecture is true, then the problem PRIMES is not in NSPACE(S(n)) for all S(n) = o(log n). However, if PRIMES is not in NSPACE(S(n)) for all S(n) = o(log n), then the strong Goldbach’s conjecture is true or this has an infinite number of counterexamples. Since Harald Helfgott proved that the weak Goldbach’s conjecture is true, then the strong Goldbach’s conjecture is true or this has an infinite number of counterexamples, where the case of infinite number of counterexamples statistically seems to be unlikely. In addition, if PRIMES is not in NSPACE(S(n)) for all S(n) = o(log n), then the Beal’s conjecture is true. Since the Beal’s conjecture is a generalization of Fermat’s Last Theorem, then this is also a simple and short proof for that Theorem.
    [Show full text]
  • A Survey of Diophantine Equations
    Part I: 1:00 PM – 2:25 PM Break: 2:30 PM – 2:40 PM Part II: 2:45 PM – 3:40 PM Modern Math Workshop 2013 Undergraduate Mini-Course #2: A Survey of Diophantine Equations Edray Herber Goins Department of Mathematics Purdue University October 2, 2013 2013 SACNAS National Conference A Survey of Diophantine Equations Part I: 1:00 PM – 2:25 PM Break: 2:30 PM – 2:40 PM Part II: 2:45 PM – 3:40 PM Abstract There are many beautiful identities involving positive integers. For example, Pythagoras knew 32 +42 =52 while Plato knew 33 +43 +53 =63. Euler discovered 594 + 1584 = 1334 + 1344, and even a famous story involving G. H. Hardy and Srinivasa Ramanujan involves 13 +123 =93 +103. But how does one find such identities? Around the third century, the Greek mathematician Diophantus of Alexandria introduced a systematic study of integer solutions to polynomial equations. In this session, we’ll focus on various types of so-called Diophantine Equations, discussing such topics as the Postage Stamp Problem, Pythagorean Triples, Pell’s Equations, Elliptic Curves, the ABC Conjecture and Fermat’s Last Theorem. 2013 SACNAS National Conference A Survey of Diophantine Equations Pythagorean Triples Part I: 1:00 PM – 2:25 PM Pell’s Equation Break: 2:30 PM – 2:40 PM Fermat’s Last Theorem and Beal’s Conjecture Part II: 2:45 PM – 3:40 PM Pythagorean Quadruples The ABC Conjecture Modern Math Workshop Undergraduate Mini-Course #2: Part I 1:00 PM – 2:25 PM SACNAS National Convention Room 210B Henry B.
    [Show full text]
  • Two Simples Proofs of Fermat 'S Last Theorem and Beal Conjecture." IOSR Journal of Mathematics (IOSR-JM) 14.6 (2018): 16-18
    Two simples proofs of Fermat ’s last theorem and Beal conjecture Mohamed Sghiar To cite this version: Mohamed Sghiar. Two simples proofs of Fermat ’s last theorem and Beal conjecture. IOSR Journal of Mathematics(IOSR-JM), International Organization Of Scientific Research, 2018, 14 (Issue 6, Ver. I), pp.16-18. hal-01918123 HAL Id: hal-01918123 https://hal.archives-ouvertes.fr/hal-01918123 Submitted on 20 Nov 2018 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. "Two simples proofs of Fermat 's last theorem and Beal conjecture." IOSR Journal of Mathematics (IOSR-JM) 14.6 (2018): 16-18. Deposited in the hal : hal-01918123. Two simples proofs of Fermat 's last theorem and Beal conjecture November 11, 2018 M. Sghiar [email protected] 9 Allée capitaine Jean Bernard Bossu, 21240, Talant, France Abstract :If after 374 years the famous theorem of Fermat-Wiles was demonstrated in 110 pages by A. Wiles [4], the puspose of this article is to give a simple demonstration and deduce a proof of the Beal conjecture. Résumé : Si après 374 ans le célèbre théorème de Fermat-Wiles a été démontré en 110 pages par A.
    [Show full text]
  • The Complexity of Mathematics Frank Vega Joysonic, Uzun Mirkova 5, Belgrade, 11000, Serbia [email protected]
    The Complexity of Mathematics Frank Vega Joysonic, Uzun Mirkova 5, Belgrade, 11000, Serbia [email protected] Abstract The strong Goldbach’s conjecture states that every even integer greater than 2 can be written as the sum of two primes. The conjecture that all odd numbers greater than 7 are the sum of three odd primes is known today as the weak Goldbach conjecture. A principal complexity class is NSPACE(S(n)) for some S(n). We show if the weak Goldbach’s conjecture is true, then the problem PRIMES is not in NSPACE(S(n)) for all S(n) = o(log n). However, if this happens, then the strong Goldbach’s conjecture is true or this has an infinite number of counterexamples. In addition, if this happens, then the Twin prime conjecture is true. Moreover, if this happens, then the Beal’s conjecture is true. Furthermore, if this happens, then the Riemann hypothesis is true. Since the weak Goldbach’s conjecture was proven, then this will certainly happen. 2012 ACM Subject Classification Theory of computation → Complexity classes; Theory of com- putation → Regular languages; Theory of computation → Problems, reductions and completeness; Mathematics of computing → Number-theoretic computations Keywords and phrases complexity classes, regular languages, reduction, number theory, conjecture, primes 1 Introduction 1.1 Goldbach’s conjecture Number theory is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions [30]. Goldbach’s conjecture is one of the most important and unsolved problems in number theory [14]. Nowadays, it is one of the open problems of Hilbert and Landau [14].
    [Show full text]
  • The ABC's of Number Theory
    The ABC's of Number Theory The Harvard community has made this article openly available. Please share how this access benefits you. Your story matters Citation Elkies, Noam D. 2007. The ABC's of number theory. The Harvard College Mathematics Review 1(1): 57-76. Citable link http://nrs.harvard.edu/urn-3:HUL.InstRepos:2793857 Terms of Use This article was downloaded from Harvard University’s DASH repository, and is made available under the terms and conditions applicable to Other Posted Material, as set forth at http:// nrs.harvard.edu/urn-3:HUL.InstRepos:dash.current.terms-of- use#LAA FACULTY FEATURE ARTICLE 6 The ABC’s of Number Theory Prof. Noam D. Elkies† Harvard University Cambridge, MA 02138 [email protected] Abstract The ABC conjecture is a central open problem in modern number theory, connecting results, techniques and questions ranging from elementary number theory and algebra to the arithmetic of elliptic curves to algebraic geometry and even to entire functions of a complex variable. The conjecture asserts that, in a precise sense that we specify later, if A, B, C are relatively prime integers such that A + B = C then A, B, C cannot all have many repeated prime factors. This expository article outlines some of the connections between this assertion and more familiar Diophantine questions, following (with the occasional scenic detour) the historical route from Pythagorean triples via Fermat’s Last Theorem to the formulation of the ABC conjecture by Masser and Oesterle.´ We then state the conjecture and give a sample of its many consequences and the few very partial results available.
    [Show full text]
  • 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.
    [Show full text]
  • The Complexity of Number Theory Frank Vega Joysonic, Uzun Mirkova 5, Belgrade, 11000, Serbia [email protected]
    Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 27 February 2020 doi:10.20944/preprints202002.0379.v2 The Complexity of Number Theory Frank Vega Joysonic, Uzun Mirkova 5, Belgrade, 11000, Serbia [email protected] Abstract The Goldbach’s conjecture has been described as the most difficult problem in the history of Mathematics. This conjecture states that every even integer greater than 2 can be written as the sum of two primes. This is known as the strong Goldbach’s conjecture. The conjecture that all odd numbers greater than 7 are the sum of three odd primes is known today as the weak Goldbach conjecture. A major complexity classes are L, NL and NSPACE(S(n)) for some S(n). Whether L = NL is a fundamental question that it is as important as it is unresolved. We show if the weak Goldbach’s conjecture is true, then the problem PRIMES is not in NSPACE(S(n)) for all S(n) = o(log n). This proof is based on the assumption that if some language belongs to NSPACE(S(n)), then the unary version of that language belongs to NSPACE(S(log n)) and vice versa. However, if PRIMES is not in NSPACE(S(n)) for all S(n) = o(log n), then the strong Goldbach’s conjecture is true or this has an infinite number of counterexamples. Since Harald Helfgott proved that the weak Goldbach’s conjecture is true, then the strong Goldbach’s conjecture is true or this has an infinite number of counterexamples, where the case of infinite number of counterexamples statistically seems to be unlikely.
    [Show full text]
  • Euler's Sum of Powers Conjecture
    Euler's sum of powers conjecture Euler's conjecture is a disproved conjecture in mathematics related to Fermat's last theorem. It was proposed by Leonhard Euler in 1769.[1] It states that for all integers n and k greater than 1, if the sum of n kth powers of positive integers is itself a kth power, then n is greater than or equal to k: k k k k a 1 + a 2 + ... + a n = b ⇒ n ≥ k The conjecture represents an attempt to generalize Fermat's last theorem, which is the special case n = 2: if k k k a 1 + a 2 = b , then 2 ≥ k. Although the conjecture holds for the case k = 3 (which follows from Fermat's last theorem for the third powers), it was disproved for k = 4, k = 5, k = 7, and k = 8. It is unknown whether the conjecture fails or holds for any value k ≥ 6. Contents Background Counterexamples Generalizations k = 3 k = 4 k = 5 k = 7 k = 8 See also References External links Background Euler was aware of the equality 594 + 1584 = 1334 + 1344 involving sums of four fourth powers; this however is not a counterexample because no term is isolated on one side of the equation. He also provided a complete solution to the four cubes problem as in Plato's number 33 + 43 + 53 = 63 or the taxicab number 1729.[2][3] The general solution of the equation is where a and b are any integers. Counterexamples Euler's conjecture was disproven by L. J. Lander and T.
    [Show full text]
  • Fermat's Last Theorem - Wikipedia, the Free Encyclopedia 17/06/13 5:22 PM Fermat's Last Theorem from Wikipedia, the Free Encyclopedia
    Fermat's Last Theorem - Wikipedia, the free encyclopedia 17/06/13 5:22 PM Fermat's Last Theorem From Wikipedia, the free encyclopedia In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive integers a, b, and c can satisfy the equation an + bn = cn for any integer value of n greater than two. This theorem was first conjectured by Pierre de Fermat in 1637, famously in the margin of a copy of Arithmetica where he claimed he had a proof that was too large to fit in the margin. No successful proof was published until 1995 despite the efforts of countless mathematicians during the 358 intervening years. The unsolved problem stimulated the development of algebraic number theory in the 19th century and the proof of the modularity theorem in the 20th century. It is among the most famous theorems in the history of mathematics and prior to its 1995 proof was in the Guinness Book of World Records for "most difficult mathematical problems". Contents 1 Overview 1.1 The problem 1.2 Subsequent developments and The 1670 edition of Diophantus' Arithmetica includes Fermat's solution commentary, particularly his "Last Theorem" (Observatio Domini 2 Mathematical History Petri de Fermat). 2.1 Pythagoras and Diophantus 2.1.1 Pythagorean triples 2.1.2 Diophantine equations 2.2 Fermat's conjecture 2.3 Proofs for specific http://en.wikipedia.org/wiki/Fermat's_Last_Theorem Page 1 of 20 Fermat's Last Theorem - Wikipedia, the free encyclopedia 17/06/13 5:22 PM exponents 2.3.1 Sophie Germain
    [Show full text]
  • Proof of Beal's Conjecture
    Vol. 11(8), pp. 109-113, November 2018 DOI: 10.5897/AJMCSR2018.0764 Article Number: F5F6E2B59269 ISSN: 2006-9731 Copyright©2018 African Journal of Mathematics and Computer Author(s) retain the copyright of this article http://www.academicjournals.org/AJMCSR Science Research Full Length Research Paper Proof of Beal’s conjecture Samuel Bonaya Buya Ngao Girls’ Secondary School, Kenya. Received 6 October 2018; Accepted 7 November, 2018 In this research, a proof of Beal’s conjecture is presented. A possible Pythagorean algebraic relationship between the terms of the conjecture problem will be proposed and used to arrive at the proof results. In the process of seeking the proof the solution of the congruent number problem through a family of cubic curves will be discussed. Key words: Proof of Beal’s conjecture, proof of ABC conjecture, algebraic proof of Fermat’s last theorem, the congruent number problem, rational points on the elliptic curve, Pythagorean triples INTRODUCTION Beal’s conjecture was formulated in 1993 by Andrew derived and used. Beal, a banker and amateur mathematician while investigating generalizations of Fermat’s last theorem. An algebraic relationship between the terms of The conjecture was formulated after some extensive Pythagorean triples computational experiments were conducted in August 1993. In some publications the conjecture has been Consider some Pythagorean integer triples x, y and z occasionally referred to as the generalized Fermat’s last related by the equation: theorem (Mauldin, 1997), (Bennet et al, 2014) the Mauldin conjecture (puzzles, n.d.) and the Tijdeman- x2 y 2 z 2 Zagier conjecture (puzzles, n.d.)], (Waldschmidt, 2004), (1) (Wikipedia, 2018).
    [Show full text]