Small two-variable exponential Diophantine equations

2004

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Using a theorem on linear forms in logarithms, we show that the equation p x À 2 y ¼ p u À 2 v has no solutions ðp; x; y; u; vÞ with xau; where p is a positive prime and x; y; u; and v are positive integers, except for four specific cases, or unless p is a Wieferich prime greater than 10 15 : More generally, we obtain a similar result for p x À q y ¼ p u À q v 40 where q is a positive prime, qc1 mod 12: We solve a question of Edgar showing there is at most one solution ðx; yÞ to p x À q y ¼ 2 h for positive primes p and q and positive integer h: Finally, we use elementary methods to show that, with a few explicitly listed exceptions, there are at most two solutions ðx; yÞ to jp x 7q y j ¼ c and at most two solutions ðx; y; zÞ to p x 7q y 72 z ¼ 0; for given positive primes p and q and integer c: r

Section: Article In Pressmentioning

confidence: 99%

On px−qy=c and related three term exponential Diophantine equations with prime bases

2004

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“…Indeed, when m = n = 1 in all solutions to (1), one can see that if x 1 < x 2 < x 3 then y 1 < y 2 < y 3 , so we may assume both x 0 2 and y 0 2 for at least one pair of solutions. 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.…”

Section: Suppose We Have Three Solutionsmentioning

confidence: 99%

The generalized Pillai equation ±rax±sby=c

2011

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In this paper we consider N, the number of solutions (x, y, u, v) to the equation (−1) u ra x + (−1) v sb y = c in positive integers x, y and integers u, v ∈ {0, 1}, for given integers a > 1, b > 1, c > 0, r > 0 and s > 0. We show that N 2 when gcd(ra, sb) = 1, except for a finite number of cases that can be found in a finite number of steps. For arbitrary gcd(ra, sb) with (u, v) = (0, 1), we show that N 3 with an infinite number of cases for which N = 3.

“…If b 0 = 1, then r 2 = 2 and r 1 = 1, so that (42) implies (26), which has been excluded. So we can take b 0 > 1.…”

Section: Note That P D a D/i P D B D/j P F A F/i And P F B F/j mentioning

confidence: 99%

“…Also, when a = b, we consider the solution (x, y) to be the same as the solution (y, x). (It is not hard to see that the existence of two solutions when a = b implies (25) or (26) below, but this is not needed here. )…”

mentioning

confidence: 99%

“…Using these facts, we proceed with a computer search: for each a < 1670, we find a prime which must divide the right side of (54), and use this prime to obtain a congruence restriction on x 2 − x 1 ; we then find a prime that must divide the left side of (54), and use it to obtain a congruence restriction on y 2 − y 1 ; continuing back and forth this way, we obtain ever stronger congruence restrictions on x 2 − x 1 and y 2 − y 1 , eventually showing that one side of (54) must be divisible by a prime in S, giving a contradiction. This is the method referred to as "bootstrapping" in [26] and is used by many authors to handle specific equations. This method cannot be relied on in general, but it does work for a < 1670, showing that there are no cases of (49a) with x 1 > m when 3 < a < 1670.…”

mentioning

confidence: 99%

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On the generalized Pillai equation ±ax±by=c

2006

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We show that the equation ±a x ± b y = c (where the ± signs are independent) has at most two solutions (x, y) for given integers a and b both greater than one and c greater than zero, except for listed specific cases. For any prime a > 5 and b = 2, we show that there are at most two values of c allowing more than one solution to this equation, not counting trivial rearrangements; further restricting a to be a non-Wieferich prime, we improve this result: we show that there are no values of c allowing more than one solution, apart from designated exceptional cases. Finally, we give all solutions to the equation |a x 1 − b y 1 | = |a x 2 − b y 2 | for b = 2 or 3 and prime a not a base-b Wieferich prime.