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Versione 1.022/3/2013Pagina 28 di 493.1 BEAL'S CONJECTUREBeal's conjecture is a conjecture that statesIfa m + b q = c kwhere a, b, c, m, n, and k are positive integers with m, n, and k ≥ 3 then a, b, c have a common factor.ExamplesTo illustrate, the solution 3 3 + 6 3 = 3 5 has bases with a common factor of 3, and the solution 7 6 + 7 7 =98 3 has bases with a common factor of 7. Indeed the equation has infinitely many solutions, includingfor examplem m m m m m m m m+1[ a ( a + b )] + [ b( a + b )] = ( a + b )for any a , b,m > 3 . But no such solution of the equation is a counterexample to the conjecture, sincethe bases all have the factorm ma + b in common.The example 7 3 + 13 2 = 2 9 shows that the conjecture is false if one of the exponents is allowed to be 2.PropertiesA variation of the conjecture where m, n, and k (instead of a, b, c) must have a common prime factor isnot true. See, for example 27 4 + 162 3 = 9 7 (3, 4 and 7 have no common factor)By the case of super-general trinomial equationPROOF
Versione 1.022/3/2013Pagina 29 di 49Aa m + Bb q = Cc kWe have Beal’s conjecture replacingA = B = C =1And so we return to the Fermat–Catalan conjecture BUT with exponent m, n, and k ≥ 3 (2 is notallowed).This shows by the new abc conjecture that we have only finitely many solutions (a,b,c,m,n,k) where(m or n or k) = 2, one of the three, otherwise the inequality doesn't hold (remember that a, b, c mustbe coprime).In fact we have trivial solutions with a common factor only with m, n, and k ≥ 3.
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