Almost perfect powers in consecutive integers

Hanrot, Guillaume; Saradha, N.; Shorey, T. N.

doi:10.4064/aa99-1-2

Cited by 16 publications

(42 citation statements)

References 16 publications

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“…Proof. We begin by noting that the stated results for equations (15), (18), (20) and (22) are, essentially, available in a paper by Bennett and Skinner [1]. The cases of equation (21) with p = 3 or 5, and β 1, while not all explicitly treated in [1], follow immediately from the arguments of that paper, upon noting that the modular curves X 0 (N ) have genus 0 for all N dividing 6 or 10.…”

Section: Proof Of Theorem 12 In Case Lmentioning

confidence: 99%

“…When Π is even, the identity {0, 2, 5, 7} yields (24) and hence a contradiction. If 9 | n + 3d, {0, 1, 6, 7} leads to (20), if Π is odd. If Π is even, from the same identity we have (24).…”

Section: The Case K =mentioning

confidence: 99%

“…By Proposition 3.1, in each case, we conclude that there is no solution to (5). Finally, if 9 | n + 6d, then the identity {0, 3, 4, 7} provides either (20) or (24). In both cases, we have a contradiction, at least for l 11 prime.…”

Section: The Case K =mentioning

confidence: 99%

“…In the special case d = 1, the set of solutions of equation (5), for k 2 fixed, has been described in [17,20,31], under less restrictive assumptions upon b. For further partial results on (5), we refer again to the survey papers [18,35,36,42].…”

Section: Introductionmentioning

confidence: 99%

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Powers from Products of Consecutive Terms in Arithmetic Progression

Bennett¹,

Bruin²,

Győry³

et al. 2006

Proceedings of London Math Soc

126

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We show that if $k$ is a positive integer, then there are, under certain technical hypotheses, only finitely many coprime positive $k$-term arithmetic progressions whose product is a perfect power. If $4 \leq k \leq 11$, we obtain the more precise conclusion that there are, in fact, no such progressions. Our proofs exploit the modularity of Galois representations corresponding to certain Frey curves, together with a variety of results, classical and modern, on solvability of ternary Diophantine equations. As a straightforward corollary of our work, we sharpen and generalize a theorem of Sander on rational points on superelliptic curves.

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Section: Proof Of Theorem 12 In Case Lmentioning

confidence: 99%

Section: The Case K =mentioning

confidence: 99%

Section: The Case K =mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Powers from Products of Consecutive Terms in Arithmetic Progression

Bennett¹,

Bruin²,

Győry³

et al. 2006

Proceedings of London Math Soc

126

View full text Add to dashboard Cite

show abstract

“…Equation (3) is a so-called binomial Thue equation, and a wide range of diophantine problems leads to such equations (see e.g. [1,2,3,11,12,14,17,18,21]). …”

Section: Reduction To the Equationmentioning

confidence: 99%