On the Diophantine Equation n(n + d) · · · (n + (k − 1)d) = by^l

Abstract

Abstract. We show that the product of four or five consecutive positive terms in arithmetic progression can never be a perfect power whenever the initial term is coprime to the common difference of the arithmetic progression. This is a generalization of the results of Euler and Obláth for the case of squares, and an extension of a theorem of Győry on three terms in arithmetic progressions. Several other results concerning the integral solutions of the equation of the title are also obtained. We extend results of Sander on the rational solutions of the equation in n, y when b = d = 1. We show that there are only finitely many solutions in n, d, b, y when k ≥ 3, l ≥ 2 are fixed and k + l > 6.

Almost fifth powers in arithmetic progression

“…, x + (k − 1)d, with gcd(x, d) = 1. The conjecture of Erdős has recently been verified for certain values of k in a more general form; see the papers [11,12,1,13]. Since now we focus on the case n = 5, we give only the best known result for this particular exponent.…”

Diophantine equations with products of consecutive values of a quadratic polynomial

“…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.…”

Rational function variant of a problem of Erdös and Graham

“…[(5, 60)] (7,9) [(1, 3) , (5, 30)] (7,11) [ (3,12) , (−6, 12)] (8,12) [(2, 6)] (10,11) [(−6, 6)] (11,14) [(1, 2)] (11,15) [(−6, 4)] (12,14) [(−7, 12)] (13,15) [(−8, 24)]…”