Number of solutions to $a^x + b^y = c^z$

Scott, Reese; Styer, Robert

doi:10.5486/pmd.2016.7282

Cited by 10 publications

(19 citation statements)

References 8 publications

(23 reference statements)

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“…Corollary 1 can also be proven without using the methods of Theorem 1 and Lemma 3, instead using the simpler methods of [16]: in this way, a result very similar to Corollary 1 is obtained in [6], which came to our attention after completion of this paper. [6] also gives a condition under which (1.3) has at most two solutions.…”

Section: Proofs Of Corollariesmentioning

confidence: 53%

“…There are many cases with N = 2 when n = 2: there are at least four infinite families of such cases, and many anomalous cases which are not members of known infinite families (the anomalous case with the largest c z which we have found is 10 5 + 41 3 = 411 2 which has the second solution 1 + 10 • 41 = 411). It seems to be a difficult problem to estimate the nature and extent of such double solutions; if one excludes from consideration cases in which min(X, Y ) = 1, then a conjecture on double solutions is given at the end of [16].…”

Section: Sharper Results For N ≤mentioning

confidence: 99%

“…noting that, just as in the treatment of (1.1) in the previous paragraph, each solution to (1.2) corresponds to a given D and a pair of ideals {c, c} in Q( √ −D). Using a generalization of the methods of [16], we show that the larger the number of D corresponding to solutions of (1.2) the smaller the number of pairs {c, c} which can occur for a given D. More precisely, we obtain a bound q on the number of pairs {c, c} which can occur with a given D, and then, letting p be the number of D corresponding to solutions of (1.2), we show that pq = 2 n−1 . The bound q is independent of the specific value of D.…”

Section: Introductionmentioning

confidence: 99%

“…In [16] this idea was used in the case n = 2 to show that there are at most two solutions in positive integers (x, y, z) to the equation a x + b y = c z where a > 1, b > 1, 2 ∤ c, improving the bound of 2 ω+1 in [9] and also improving the absolute bound of 2 36 obtained by Hirata-Kohno [8] using the non-elementary work of Beukers and Schlickewei cited above [1] (there are an infinite number of (a, b, c) giving exactly two solutions).…”

Section: Introductionmentioning

confidence: 99%

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Two terms with known prime divisors adding to a power

Scott¹,

Styer²

2018

Publ. Math. Debrecen

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Let c be a positive odd integer and R a set of n primes coprime with c. We consider equations X + Y = c z in three integer unknowns X, Y , z, where z > 0, Y > X > 0, and the primes dividing XY are precisely those in R. We consider N , the number of solutions of such an equation. Given a solution (X, Y, z), let D be the least positive integer such that (XY /D) 1/2 is an integer. Further, let ω be the number of distinct primes dividing c. Standard elementary approaches use an upper bound of 2 n for the number of possible D, and an upper bound of 2 ω−1 for the number of ideal factorizations of c in the field Q( √ −D) which can correspond (in a standard designated way) to a solution in which (XY /D) 1/2 ∈ Z, and obtain N ≤ 2 n+ω−1 . Here we improve this by finding an inverse proportionality relationship between a bound on the number of D which can occur in solutions and a bound (independent of D) on the number of ideal factorizations of c which can correspond to solutions for a given D. We obtain N ≤ 2 n−1 + 1. The bound is precise for n < 4: there are several cases with exactly 2 n−1 + 1 solutions.where c is a fixed positive odd integer, gcd(XY, c) = 1, and the set of primes in the factorization of XY is prechosen. Previous work on this problem includes both strictly elementary treatments and deeper, non-elementary approaches.The most common type of non-elementary approach uses results on S-unit equations to obtain a bound which is exponential in s, where s is the number of primes dividing XY c. A familiar general result of Evertse [7] shows that there are at most exp(4s + 6) solutions to the equation x + y = z in coprime positive integers (x, y, z) each composed of primes from a given set of s primes. It follows Mathematics Subject Classification: 11D61.

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Section: Proofs Of Corollariesmentioning

confidence: 53%

Section: Sharper Results For N ≤mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Two terms with known prime divisors adding to a power

Scott¹,

Styer²

2018

Publ. Math. Debrecen

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“…(ii) If max{a, b, c} > 5 × 10 27 , then N(a, b, c) ≤ 3. R. Scott and R. Styer [9] proved that if 2 ∤ c, then N(a, b, c) ≤ 2. The proofs of the first two results are using the Gel'fond-Baker method with an elementary approach, and the proof of the last result is using some elementary algebraic number theory methods.…”

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confidence: 99%

An upper bound for the number of solutions of ternary purely exponential Diophantine equations II

Hu¹,

Mao-hua²

2019

Publ. Math. Debrecen

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Let a, b, c be fixed coprime positive integers with min{a, b, c} > 1. In this paper, by analyzing the gap rule for solutions of the ternary purely exponential diophantine equation a x + b y = c z , we prove that if max{a, b, c} ≥ 10 62 , then the equation has at most two positive integer solutions (x, y, z).(x, y, z) of (1.1). As a straightforward consequence of an upper bound for the number of solutions of binary S−unit equations due to F. Beukers and H. P. Schlickewei [2], we have N(a, b, c) ≤ 2 36 . In nearly two decades, many papers investigated the exact values of N(a, b, c). The known results showed that (1.1) has only a few solutions for some special cases(see the references of [5] and [6] ).Recently, Y.-Z. Hu and M.-H. Le [5,6] successively proved that (i) if a, b, c satisfy certain divisibility conditions and max{a, b, c} is large enough, then (1.1) has at most one solution (x, y, z) with min{x, y, z} > 1. (ii) If max{a, b, c} > 5 × 10 27 , then N(a, b, c) ≤ 3. R. Scott and R. Styer [9] proved that if 2 ∤ c, then N(a, b, c) ≤ 2. The proofs of the first two results are using the Gel'fond-Baker method with an elementary approach, and the proof of the last result is using some elementary algebraic number theory methods. In this paper, by analyzing the gap rule for solutions of (1.1) along the approach given in [6], we prove a general result as follows:Notice that, for any positive integer k with k > 1, if (a, b, c) = (2, 2 k − 1, 2 k + 1), then (1.1) has solutions (x, y, z) = (1, 1, 1) and (k + 2, 2, 2). It implies that there exist infinitely many triples (a, b, c) which make N(a, b, c) = 2. Therefore, in general, N(a, b, c) ≤ 2 should be the best upper bound for N(a, b, c). PreliminariesLemma 2.1 Let t be a real number. If t ≥ 10 62 , then t > 6500 6 (log t) 18 .Proof. Let F (t) = t − 6500 6 (log t) 18 . Then we have F ′ (t) = 1 − 18 × 6500 6 (log t) 17 /t and F ′′ (t) = 18×6500 6 (log t) 16 (log t−17)/t 2 , where F ′ (t) and F ′′ (t) are the derivative and the second derivative of F (t). Since F ′ (10 62 ) > 0 and F ′′ (t) > 0 for t ≥ 10 62 , we get F ′ (t) > 0 for t ≥ 10 62 . Further, since F (10 62 ) > 0, we obtain F (t) > 0 for t ≥ 10 62 . The lemma is proved.Let α be a fixed positive irrational number, and let α = [a 0 , a 1 , . . . ] denote the simple continuous fraction of α. For any nonnegative integer i, let p i /q i be the i−th convergent of α. By Chapter 10 of [7], we obtain the following two lemmas immediately.Lemma 2.2 (i) The convergents p i /q i (i = 0, 1, . . . ) satisfy p −1 = 1, p 0 = a 0 , p i+1 = a i+1 p i + p i−1 , q −1 = 0, q 0 = 1, q i+1 = a i+1 q i + q i−1 , i ≥ 0.

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