The Surprising Proofs

Transcription

1 The Surprising Proofs By Leszek W. Guła Lublin-POLAND May 30, June and July 02 03, 2017 Abstract. The proof of the Fermat s Last Theorem. The proof of the theorem - For all and for all and for all natural numbers The proof of the Goldbach s Conjecture. The proof of the Beal s Conjecture. MSC. Primary: 11A41, 11D41, 11P32; Secondary: 11A51, 11D45, 11D61. Keywords. Algebra of Sets, Diophantine Equations, Exponential Equations, Fermat Equation, Goldbach Conjecture, Greatest Common Divisor, Newton Binomial Formula. 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. The Goldbach s Conjecture is one of the oldest and best-known unsolved problems in number theory and all of mathematics. It states: Every even integer greater than 2 can be expressed as the sum of two primes. Let and be positive integers with If then and have a common factor. Beal s conjecture is a generalization of Fermat s Last Theorem. It states: If, where and are positive integers and and are all greater than, then and must have a common prime factor. Or the below - slightly restated? The Beal Prize Fund is with US$1,000,000 to be awarded in a case presented a counterexample. II. THE PROOF OF THE FERMAT S LAST THEOREM For all Suppose that for some and for all Then otherwise Thus for some

2 At present we can assume for generality of below that and are coprime. Moreover Every even number which is not the power of number has odd prime divisor, hence sufficient that we prove FLT for and for odd prime numbers Without loss for the proof we can assume that is even. For some Notice that For some coprime Moreover For some pairwise (mutually) relatively prime such that Therefore For some relatively prime such that inasmuch as on the strength of the Guła s Theorem we have This is the proof. Without loss for the proof we assume that: is odd and The numbers and are coprime, therefore in view of we will have For some

3 We assume that For some such that and are coprime if if

4 if if if if For some and for some such that and and are coprime and

5 This is the proof. III. PROOF OF - If and then for all For all and for all the equation has no primitive solutions in Proof. Suppose that for some the equation has primitive solutions in Then and are coprime and odd Without loss for the proof we can assume that On the strength of the Guła s Theorem we get inasmuch as and because the numbers are odd or are odd. where the numbers are positive and odd and This is the proof. Golden Nyambuya proved (allegedly) the theorem For all the equation has no primitive solutions in with Corollary 1. For some prime natural numbers such that is positive and odd This is the Corollary 1.

6 Example 1. where and This is the Example 1. Example 2. where and This is the Example 2. Example 3. where and This is the Example 3. Example 4. where and This is the Example 4. Example 5. where and This is the Example 5. such that For all and for all This is the theorem. For all and for all such that the number is even and bigger than two and This is the theorem.

7 IV. THE PROOF OF THE GOLDBACH S CONJECTURE For all Thus Hence whence it implies that for all This is the proof. V. THE PROOF OF THE BEAL S CONJECTURE For all the equation has no primitive solutions in

8 Suppose that for some the equation has primitive solutions in Without loss for the proof we can assume that Then only one number out of is even and and where the positive natural numbers are coprime and is odd, and or because the Jeśmanowicz Conjecture is true. Therefore we will have Hence For some such that is odd and are coprime Since then we will have We assmume that the number is minimal. For some such that and are coprime minimal

9 If the bases in three powers we define as conclusions of the above implications that is to say from the above conditions, then we will not find such the arithmetic triple (A, B, C) for which our equation will be false. Yes that's is in the proof of FLT for odd, so the Beal's Conjecture is true. This is the proof. For each fixed pair of the relatively prime natural numbers and such that is positive and odd there exists exactly one a primitive Pythagorean triple such that and conversely Any the primitive Pythagorean triple such that arises exactly from one pair of the relatively prime natural numbers and such that is positive and odd. This is the theorem. Let and Suppose that for some such that the numbers and are coprime and We assume that the number is minimal. On the strength of the Guła s Theorem we obtain minimal If then on the strength of the Guła s Theorem we get It s not true in that FLT for can be written equivalently as: where or because Fermat did not proved his own theorem for The below we have the new proofs in the above two cases. In the first case we will have If then on the strength of the Guła s Theorem we get

10 In the second case we have. AMS: En. Wikipedia Guła, L. W.: Disproof the Birch and Swinnerton-Dyer Conjecture, American Journal of Educational Research, Volume 4, No 7, 2016, pp , doi: /education Original Article electronically published on May 3, Guła, L. W.: Several Treasures of the Queen of Mathematics, International Journal of Emerging Technology and Advanced Engineering, Volume 6, Issue 1, January 2016, pp Guła, L. W.: The Truly Marvellous Proof, International Journal of Emerging Technology and Advanced Engineering, Volume 2, Issue 12, December 2012, pp Mauldin, R. Daniel: A Generalization of Fermat s Last Theorem: The Beal Conjecture and Prize Problem, Notices of the AMS,, Volume 44, No 11, December 1997, p Mazur, B.: Mathematical Perspectivies, Bulletin (New Series) of The American Mathematical Society, Volume 43, Number 3, July 2006, p. 399, Article electronically published on May 9, Narkiewicz, W.: Wiadomości Matematyczne XXX.1, Annuals PTM, Series II, Warszawa 1993, p. 3. Nyambuya, G. G.: On a Simpler, Much More General and Truly Marvellous Proof of Fermat s Last Theprem (II), Department of Applied Physics, National University of Science and Technology, Bulawayo, Republic of Zimbabwe, Preprint submitted to vixra.org Version 3 - September 24, 2014, pp 6-12,