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). x2 a The conjecture states that: a If y (2) x y z 2 If ABC, where A, B, C are positive integers and x, y and z are all positive integers greater 2, then A, B x2 and C have a common prime factor. a And again z a (3) To prove the conjecture a Pythagorean algebraic 2 relationship between the terms of the conjecture will be
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y2 x 3 n 2 x (3), Then by Equations 2 and 3 where n is congruent number if n is of the form z y a (4) b32 a b p n a, b p , q N q 0 4aq x2 Consider the elliptic curve 3. Making the congruent zy (5) a number (n) the subject of the equation we obtain the following: Multiplying Equations 5 and 4 together: 2 y y y n22 x x x (4) xx22 aa x xx x2( z y )( z y ) z 2 y 2 (aa ) 2 ( ) 2 22 xx22 For n to be a congruent number take: aa x2()()aa 2 2 (6) y 2 22 xa (5) x A relationship has been established between the terms of y1 a b2 a Pythagorean triple. Some forms of the above x ()2 Pythagorean relationships were first discussed in a paper 42b (6) entitled simple algebraic proofs of Fermat’s last theorem x (Buya, 2017a) From Equations 5 and 6
Solution of the congruent number problem 2 1122ab The technique used in proving Beal conjecture can be xa(()) (7) used to solve problems in number theory, arithmetic of 2 4 2b elliptic of elliptic curves etc. (Elkies, 2007). An important problem in number theory that can be solved is the 22 12 1a b 2 1 2 1 a b 2 congruent number theory (Buya, 2017b). y( a ( ) )( ( a ( ) )) (8) The results in section 2 can be used to solve the 4 4 2bb 2 4 2 congruent number problem. If the perpendicular height of a right angled triangle is x then by equation 7 its base is For a rational number on the elliptic curve 3 select x2 rational numbers a and b so that x is a square rational a a number. The family of cubic curves 2 can also be used to . The area of a congruent triangle is given by: establish whether the elliptic curve 3 has or does not 2 have a rational number.
2 x a x32 a x Ax()a (1) Proof of Beal’s conjecture 44a Relationship between the terms of Beal’s conjecture For the purpose of solution of the congruent number problem (a) is considered to be a rational number in this In a paper entitled simple algebraic proofs of Fermat’s section. last theorem (Buya, 2017a) the author of this paper Thus the family of curves of the cubic equation showed that there exists an algebraic relationship between the terms of Fermat’s Diophantine equation. The x32 a x y (2) algebraic equations in the paper can be used to prove 4a Beal conjecture. There exists an algebraic relationship has an infinite number of points for solution of the between the terms of Beal’s conjecture problem. The congruent number problem. In the function 2 above the y proposed proof does not require any Galois coordinate represents the area of a congruent number representation or use of elliptic curves. with (x) and (a) as rational numbers. Thus in the elliptic curve: Consider the equation given by: Buya 111
x Ax B y C z x, y , z 2 (1) A 1 2x x/ z 2/ z mv()m (10) The terms of the Equation above are related in the 2 following way: Substituting equations 9 and 10 into 5 and 6: Ax a2 a x 1 A x xy/ By ()()()a 2 2A 2 (2) B m u (11) 22a C mxz/ v (12) Ax a2 a x 1 x The set of equations 7, 11 and 12 suggest that A, B and z a 2A 2A 2 C ()()() (3) C at least share a common prime number validating 22a Beal’s conjecture. To prove the Beal’s conjecture, the cases Example 1 Ax ax a, m (4) In Equations 7, 11 and 12 if m = 16, x = 3 y = 4 z = 3 m then: will be considered.
Substituting 4 into 2 and 3:
AAxx a 1 2x 2x x 2 x x 2 2 x x 2 yam2 x 2 2 2 x m A 2 ()() m A m am B()()()() m m 2x 2 x 2 x 2 2 2m 4 m 4 m 3 3 4 3 4 x 16a 16 u 16 v A 1 2x 3 4 3 4 2x / y 2/ y 16 (a u ) 16 v Bm()m (5) 2 4 AAxx If we take v = 2, a = 8 and u = 8 we obtain the Equation: a 1 2x x 2 x x 2 2x m A() m am Cz()()()()()am2 m x 2 2 m x 2 2 22xx 3 3 4 2 2 2mm 4 32 32 16
x A The common prime factor in this case is 2. 1 2x 2x / zm 2/ z Cm() (6) Example 2 2 In Equations 7, 11 and 12 if m = 3, x = 6 y = 3 and z = 2 From Equation 4: then:
xx A am Aax 36 (7) 1/ x y 63 A a m Bu 3 Cvz 382 Substituting 4, 5 and 6 into 1: 36 (a u 3 ) 3 8 v 2 2x x 2 2 x x 2 x ()()m am m am am 22xx (8) If we take v = 1, a = 1 and u = 2 we end up getting the 44mm identity:
For validity of Beal’s conjecture if: 6 3 8 3 18 3
x A The common prime factor in this example is 3. Thus the 1 2x mux/ y()m 2/ y (9) above derived relationships show that whenever the 2 powers of A, B and C are greater than 2 then the Beal’s 112 Afr. J. Math. Comput. Sci. Res.
conjecture identity will always have a common prime Then on substituting 18 into 15: factor shared by A, B and C. Thus, Beal’s conjecture is C proved. The number may or may not be co-primed with A. The To completely validate the conjecture other cases will be Beal’s conjecture is therefore not applicable to brought into consideration. Pythagorean Diophantine equations. The case x = y = z = 2 will be considered. This is to say we will consider the Pythagorean Diophantine equation. The case x = 3, y = z = 2 will be considered. In such a case, the algebraic relationships between A, B and C is given by: Consider the general Diophantine equation:
A2 ABC3 2 2 (19) a A221 a A Ba ( ) a A a ^ r N (13) 22a A2 r The relationship between the terms of the above Equation will be considered. Equation 13 can be further simplified. A3/2 a 3/2 2 2 Aa1 A B a () (20) a 11 A 22aA3/2 Ba Ar( ) ( r2 2) (14) 2 2rr2 2 A3/2 a 3/2 2 2 a Aa1 (21) A CB () 3/2 a 22 22aA a A1 a 1 1 A 2 (15) C ( 22 ) Ar ( ) ( r 2) 2a 2 A 2 r 2 r If in Equations 21 and 22:
Substituting 14 and 15 into 1: 1 aa2 () (22) AA22 2 AA3/2 3/2 aa 1 aa2 A2()()aa 2 2 22 (16) 2 AA 3/2 3/2 (23)
2 2 21 1 2 2 2 1 1 2 , N A A r()() 22 A r 22rr Then by the Equations 23 and 24: r2 2 Consider the algebraic fraction of equation 14. If 2r BC it has a form: The number B may or may not be co-primed with A. r2 2 The number C may or may not be co-primed with A. N (17) 2rA Thus Beal’s conjecture does not apply to the cases ( ) = ( ) and ( ) = ( )
Then on substituting equation 17 into 14: The case ( ) = ( ) will be considered. B In this case the relationship between A, B and C is given by the set of equations below: The number may or may not be co-primed with A A2 a 22 2 Aa r 2 B a (1 ) (24) Consider the algebraic fraction of equation 15. If 22aA2 2r it has a form: A2 2 a Aa22 r 2 C3/2 a (1 ) (25) 22aA2 2rA (18) Aa32 C (1 )3/2 N (2aA )3/2 2 Buya 113
REFERENCES If in Equations 24 and 25: Bennet MA, Chen I, Dahmen SR, Yazdani S (2014). Generalized Fermat Equations: A Miscellany. Simon Fraser University. Not cited aa2 2 Buya SB (2017a). Simple Algebraic proofs of Fermat's last theorem. 1 Advances in Applied Science Research 8(3): 60:64. AA2 (26) Buya SB (2017b). Solution of the Congruent number problem and proof of Birch Swinnerton-Dyer conjecture. Academia.edu. N Elkies ND (2007). The ABC's of Number theory. The Havard College aa2(2 ) 3/2 mathematics Review 1(1). (1 )3/2 Mauldin RD (1997). A Generalization of Fermat's last theorem: Beal’s 23 (27) conjecture and the Prize Problem. Notices of The AMS 44(11):1436- AA 1437. N Puzzles (n.d.). Problems and puzzles. Retrieved September 26th, 2018, from Puzzle 559 - Maudin/Tijman-Zagier conjecture: http:llwww.primepuzzles.net/puzzles/puzz_559.htm Not cited Then: Waldschmidt M (2004). Open Diophantine problems. Moscow mathematics 4:245-305. Wikipedia (2018). Retrieved from Beal’s conjecture: BC https://en.m.wikipedia.org/wiki/Beal-conjecture
In the case above C may or may not be co-primed to A. The Beal’s conjecture does not apply to the general case ( ) = ( ).
From the above analysis we note that in cases where x, y and z are greater than two, A, B and C share a common prime number. Thus Beal’s conjecture is thus verified.
Conclusion
There exists an algebraic relationship connecting the terms of Beal’s conjecture problem. The algebraic relationship as stipulated in equations 7, 11 and 12 of section 4 shows that the terms A, B and C share a common prime factor for all x, y and z as positive integers greater than 2. Thus Beal’s conjecture is proved.
CONFLICT OF INTERESTS
The author has not declared any conflict of interests.