Beal Conjecture Original Directly Proved Abstract

Transcription

1 Beal Conjecture Original Directl Proved "% of the eole think; 0% of the eole think that the think; and the other 8%would rather die than think."----thomas Edison "The simlest solution is usuall the best solution"---albert Einstein Abstract Using a direct construction aroach, the author roves the original Beal conjecture that if Ax + B C, where ABCx,,,,, are ositive integers and x,, > 2, then A, Band C have a common rime factor. Two main tes of equations are involved, namel, the equation Ax + B C and an equation which will be called a tester equation. A tester equation has similar roerties as Ax + B C, and will be used to determine the roerties of Ax + B C. Also, two tes of tester equations, namel, a literal tester equation and a numerical tester equation will be alied. Each side of Ax + B C and a tester equation is reduced to unit b division. The non-unit sides are justifiabl equated to each other to roduce a new equation which will be called the master equation. The side of the master equation involving the terms of the tester equation will be called the tester side of the master equation. Three versions of the roof are resented. In Version roof, the tester equation was the literal equation Gm + Hn I, but in Versions 2 and roofs, the tester equations were the numerical tester equations, and + 6, resectivel. B insection, using an aroach in which the corresonding elements on the right and left sides of the master equation are equated to each other, it is determined that if Ax + B C, A, Band C have a common rime factor. The roof is ver simle, and occuies a single age, and high school students can learn it.

2 Beal Conjecture Original Directl Proved (Version ) (Proof using a literal tester equation) In this version, using a literal tester equation, the author roves directl the original Beal conjecture that if Ax + B C, where ABCx,,,,, are ositive integers and x,, > 2, then A, Band C have a common rime factor. Given: Ax + B C, where A, B, C, x,, are ositive integers and x,, > 2. Required: To rove that A, B, and C, have a common rime factor. Let G, H, I, m, n, be ositive integers such that m, n, > 2, and Gm + Hn I is true. The equation Gm + Hn I will be called the tester equation (a true equation having similar roerties as Ax + B C and which will be used to determine the roerties of Ax + B C). Consider the true equations Ax + B C (); and Gm + Hn I From (), one obtains A x + B C (); and similarl, from (2), one obtains G m + H n I (4) (Dividing both sides of each equation b the right side) Let r, s, t be rime factors of A, B and C resectivel such that A Dr, B Es, and C Ft, where D, E, F are ositive integers. Also, let h, q, w be rime factors of G, H and I resectivel such that G Jh, H Kq, and I Lw, where J, K, L are ositive integers, and h q w. Then equations () and (4) become and ( Jh ) m + ( Kq ) n Examle on the rincile alied below, resectivel. Since ( Dr ) x + ( Es ) and ( Jh ) m + ( Kq ) n If U + V + U, V, and W 8 or ( Jh) m + ( Kq) n (6) (Master equation) U, V, and W 8 (B the transitive equalit roert) U + V One will next show that r s t b insection, + + using a corresondence aroach in which the ( Jh) m,( Es) ( Kq) n,, or corresonding elements on the right and left sides of equation (6) are equated to each other. ( Kq) n,, ( Es) ( Jh) m,, A Dr Jh; x m; Note: D, E, F, r, s, t, J, K, L h, q, w Es Kq, n; Ft Lw, D J; E K; F L are all integers. x,, > 2; m, n, > 2. r h; s q; t w. h q w From above, r h, s q, t w. Since h q w (given), r s t. Therefore, Dr, Es, and Ft have a common rime factor. Also, since A Dr, B Es, and C Ft, A, B, and C, have a common rime factor..or B Dr Kq; x n; Es Jh, m; Ft Lw, Note: D, E, F, r, s, t, J, K, L h, q, w D K; E J; F L are all integers. x,, > 2; m, n, > 2. r q; s h; t w. h q w From above, r q, s h, t w. Since h q w (given), r s t. Therefore, Dr, Es, and Ft have a common rime factor. Also, since A Dr, B Es, and C Ft, A, B, and C have a common rime factor. Observe above that in Table A or Table B, it is shown that r s t; and therefore, if Ax + B C, then A, B, and C have a common rime factor. The roof is comlete. 2

3 Beal Conjecture Original Directl Proved (Version 2) (Proof using a numerical tester equation) In this version, using a numerical tester equation, the author roves the original Beal conjecture that if Ax + B C, where ABCx,,,,, are ositive integers and x,, > 2, then A, Band C have a common rime factor. Given: Ax + B C, where A, B, C, x,, be ositive integers such that x,, > 2. Required: To rove that A, B, and C, have a common rime factor. Consider the true equations Ax + B C (); and Equation (2) will be called a tester equation (a true equation having similar roerties as Ax + B C and which will be used to determine the roerties of Ax + B C). From (), one obtains A x + B C (); and similarl, from (2), one obtains (4.) (Dividing both sides of each equation b the right side) Let r, s, t be rime factors of A, B and C resectivel such that A Dr, B Es, and C Ft, where D, E, F are ositive integers. Then equation () becomes (). Since ( Dr ) x + ( Es ), and Tester 678side If U + V + (B the transitive equalit roert) U, V, and W 8 or U, V, and W 8 ( 2 ) 9 + ( 2 4) ( 2 2) (6) (Master equation) U + V + + (Note: D, E, F, r, s, t, are all integers; x,, > 2) One will next show that r s t b insection, using a corresondence aroach in which the corresonding elements on the right and left sides of equation (6) are equated to each other. A A Dr 2, x 9 B Es 8 2 4, D E 4 r 2 s 2 From above, r 2, s 2, t 2, and therefore, r s t rime factor, 2. Since A Dr B Es C Ft B A Dr 2 4, x B Es 2 2, 9 D 4 E r 2 s 2 From above, r 2, s 2, t 2, and therefore, r s t rime factor, 2. Since A Dr B Es C Ft Note: Examle on the rincile alied below x 9 ( Dr) 2,( Es) 8, 4, or 8, ( Es) 29, C Ft 4 2 2, F 2 t 2, and Dr, Es, and Ft have a common,, and, A, B, and C have a common rime factor; OR C Ft 4 2 2, F 2 t 2, and Dr, Es, and Ft have a common,, and, A, B, and C have a common rime factor. Observe above that in either Table A or Table B, it is shown that r s t; and therefore, if Ax + B C, then A, B, and C, have a common rime factor. The roof is comlete.

4 Beal Conjecture Original Directl Proved (Version ) (Proof using a numerical tester equation) In this version, using a numerical tester equation, the author roves the original Beal conjecture that if Ax + B C, where ABCx,,,,, are ositive integers and x,, > 2, then A, Band C have a common rime factor. Given: Ax + B C, where A, B, C, x,, be ositive integers such that x,, > 2. Required: To rove that A, B, and C, have a common rime factor. Consider the true equations Ax + B C (); and + 6 Equation (2) will be called a tester equation (a true equation having similar roerties as Ax + B C and which will be used to determine the roerties of Ax + B C). From (), one obtains A x + B C (); and similarl, from (2), one obtains + 6 (4.) (Dividing both sides of each equation b the right side) Let r, s, t be rime factors of A, B and C resectivel such that A Dr, B Es, and C Ft, where D, E, F are ositive integers. Then equation () becomes (). Note: Since ( Dr ) x + ( Es ), and 6 Examle on the rincile alied below + Tester 678side + 6 If U + V + (B the transitive equalit roert) U, V, and W 8 or U, V, and W 8 ( ) + ( 2) ( ) (6) (Master equation) U + V + + (Note: D, E, F, r, s, t, are all integers; x,, > 2) One will next show that r s t b insection, using, ( Es) 6,, or a corresondence aroach in which the corresonding elements on the right and left sides of equation (6) 6, ( Es), are equated to each other. A A Dr, x B Es 6 2, C Ft, D E 2 F r s t From above, r, s, t, and therefore, r s t, and Dr, Es, and Ft have a common rime factor,. Since A Dr, B Es, and C Ft, A, B, and C have a common rime factor; OR B A Dr 6 2, x B Es, C Ft, D 2 E F r s t From above, r, s, t, and therefore, r s t, and Dr, Es, and Ft have a common rime factor,. Since A Dr, B Es, and C Ft, A, B, and C have a common rime factor. Observe above that in either Table A or Table B, it is shown that r s t; and therefore, if Ax + B C, then A, B, and C, have a common rime factor. The roof is comlete. 4

5 Conclusion Using a direct construction aroach, the author roved the original Beal conjecture that if Ax + B C, where ABCx,,,,, are ositive integers and x,, > 2, then A, Band C have a common rime factor. Two main tes of equations were involved, namel, the equation Ax + B C and an equation which was called a tester equation. A tester equation has similar roerties as Ax + B C, and was used to determine the roerties of Ax + B C. Two tes of tester equations, namel, a literal tester equation and a numerical tester equation were alied. Each side of Ax + B C and a tester equation was reduced to unit b division. The non-unit sides were justifiabl equated to each other to roduce a new equation which was called the master equation. The side of the master equation involving the terms of the tester equation was called the tester side of the master equation. Three versions of the roof were resented. In Version roof, tester equation was the literal equation Gm + Hn I, but in Versions 2 and roofs, the tester equations were the numerical tester equations, and + 6, resectivel.. B insection, using an aroach in which the corresonding elements on the right and left sides of the master equation are equated to each other. An interesting observation was that it did not matter, so far as the determination of the common factors were concerned, whether the elements in the first term of the numerator on the left side of the master equation were equated to the elements in either the first or second term of the numerator on the right side of the master equation. The common factor results were alwas the same. It was determined that if Ax + B C, then A, Band C have a common rime factor. The roof is ver simle, and occuies a single age, and even high school students can learn it. PS Previousl, the author roved the equivalent Beal conjecture: vixra: Adonten