Small Two-Variable Exponential Diophantine Equations

Abstract

Abstract. We examine exponential Diophantine equations of the form abx = cd? +e. Consider a < 50, c < 50, \e\ < 1000, and b and d from the set of primes 2,3,5,7, 11, and 13. Our work proves that no equation with parameters in these ranges can have solutions with x > 18. Our algorithm formalizes and extends a method used by Guy, Lacampagne, and Selfridge in 1987.

The generalized Pillai equation $\pm r a^x \pm s b^y = c$

“…For each choice of (r, a, s, b) we use the technique known as 'bootstrapping' (see [7] and [19]) to find increasingly stringent congruence conditions on the exponents x h − x 0 and y h − y 0 . When these conditions show that either x h − x 0 or y h − y 0 exceeds 8 • 10 14 , by Theorem 1 there can be no third solution.…”

“…When these conditions show that either x h − x 0 or y h − y 0 exceeds 8 • 10 14 , by Theorem 1 there can be no third solution. (The bootstrapping in [7] and [19] deals only with the case m = n = 1 but the ideas extend easily to the other choices of signs, indeed, more easily since for m = 0 or n = 0 parity considerations often lead to a contradiction, as in the proof of Lemma 9 of [17]. The Maple programs used can be found on the second author's website [20].)…”

“…Note that both sides of (19) equal the integer T /t. By the same argument as above, if there exists a fourth solution (x 4 , y 4 ), we have a solution to (19) for the same values of A and B but with m = x4−x1 x2−x1 and n = y4−y1 y2−y1 , contradicting Theorem 1.3 of [8], which states that, for given A and B, (19) has at most one solution (m, n).…”

The number of solutions to the generalized Pillai equation \pm r a^x \pm s b^y = c.

“…In this paper we consider N , the number of solutions (x, y, u, v) to the equation [1], [3], [5], [6], [15], [17], [18]). In [15] we treated (1.1) with various additional restrictions on x, y, u, v, a, b.…”

“…In the case gcd(a, b) > 1, Theorem 1.1 completely designates all exceptions. In the case gcd(a, b) = 1, since any basic form for which gcd(a, b) = 1 must also satisfy gcd(ra, sb) = 1, Theorem 1.2 reduces the problem to a finite search for basic forms; another (somewhat lengthy) paper [16] completes the search using not only the methods of [4] and [18] as in previous work by the authors but also using LLL basis reduction. The completion of the search in [16] proves Theorem A below, for which we need one more definition:…”