Abstract:We prove that for any positive integers x, d and k with gcd(x, d) = 1 and 3 < k < 35, the product x(x + d) · · · (x + (k − 1)d) cannot be a perfect power. This yields a considerable extension of previous results of Győry et al. and Bennett et al., which covered the cases where k ≤ 11. We also establish more general theorems for the case where x can also be a negative integer and where the product yields an almost perfect power. As in the proofs of the earlier theorems, for fixed k we reduce the problem to syst… Show more
“…For small values of k, finiteness results for (1.1) (under coprimality assumptions) have been obtained for k ≤ 82 in [1,Theorem 1.4] and, in completely explicit form for k ≤ 34, in [10]. The techniques of [1] do not allow for substantial strengthening of these results, however.…”
Section: Introduction a Striking Results Of Erdős And Selfridgementioning
“…For small values of k, finiteness results for (1.1) (under coprimality assumptions) have been obtained for k ≤ 82 in [1,Theorem 1.4] and, in completely explicit form for k ≤ 34, in [10]. The techniques of [1] do not allow for substantial strengthening of these results, however.…”
Section: Introduction a Striking Results Of Erdős And Selfridgementioning
“…One can cite for examples [4,[8][9][10][11][12][13]16,17,20,21]. In particular, Laishram and Shorey [16,17], Győry [11], Győry et al [12,13], and Bennett et al [4] give several finiteness results concerning the solutions of Eq. (2) under some assumptions.…”
Available online xxxx Communicated by Michael E. Pohst MSC: primary 11D09 secondary 11B85 Keywords: Diophantine equation Irreducible quadratic polynomials Chebyshev's inequality Let a, b, c, d be given nonnegative integers with a, d 1. Using Chebyshev's inequalities for the function π (x) and some results concerning arithmetic progressions of prime numbers, we study the Diophantine equation n k=1ak 2 + bk + c = dy l , gcd(a, b, c) = 1, l 2,where ax 2 + bx + c is an irreducible quadratic polynomial. We provide a computable sharp upper bound to n. Using this bound, we entirely prove some conjectures due to Amdeberhan, Medina and Moll (2008) [1]. Moreover, we obtain all the positive integer solutions of some related equations.
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