On the Number of Solutions to the Generalized Fermat Equation

We discuss the maximum number of distinct non-trivial solutions that a generalized Fermat equation Ax n + By n = Cz n might possibly have. The abc-conjecture implies that it can never have more than two solutions once n > n 0 (independent of A; B;C); and that it has no solutions for xed A; B; C once n > n A;B;C. On the other hand for any set of pairwise coprime integers p 1 ; p 2 ; :: : ; p k , no matter how large, we will construct non-zero integers A; B; C such that there are distinct non-trivial solutions to Ax p i i +By p i i = Cz p i i for i = 1; 2; : :: ; k. We also show that n 0 > 4. In the nal section we review some recent relevant results, consider generalizing these questions to all curves, and also brieey discuss the challenge of modifying the Frey{Ribet application of the Taniyama-Shimura-Weil Conjecture to Fermat's Last Theorem, to attack the generalized Fermat equation. In about 1637 Fermat made the assertion, popularly known today as Fermat's Last Theorem 1 , that there are no non-trivial integer solutions x; y; z; p 3 to the equation x p + y p = z p. In this paper we are interested in the number of non-trivial coprime integer solutions x; y; z, with p 4, to the more general Fermat equation (1) Ax p + By p = Cz p ; where A; B; C are given non-zero integers with gcd(A; B; C) = 1. We note here that this is equivalent to counting rational points on the curve C ;;;p : u p + v p = 1: This equivalence may be seen by`de-homogenizing' (1) with the transformation = A=C; = B=C; u = x=z and v = y=z, and`homogenizing' C ;;;p by multiplying points on this curve through by an appropriate common denominator. We missed out the exponents p = 2 and 3 in our formulation of this problem for the same reason that Fermat missed out p = 2 in his problem. That is that, for these p, there are usually innnitely many solutions to (1) if there is one. Indeed if p = 2 then we consider the line of rational slope t that goes through the given point on the curve C ;;;2 , …