Almost perfect powers in arithmetic progression

Abstract

1. Introduction. Let b, d, , n and y be positive integers such that gcd(n, d) = 1 and > 2 is a prime number. Let k ≥ 2, t ≥ 2 and r ∈ {0, 1} be integers satisfying t = k − r. Thus k ≥ 2 if r = 0 and k ≥ 3 if r = 1. Let d 1 < . . . < d t be integers in the interval [0, k). We write

The Least Common Multiple of Consecutive Terms in a Quadratic Progression

“…See [2,15] for some other results. However, there are few renowned theorems but more conjectures about quadratic progressions, among which the sequence {n 2 + 1} n∈N is best known.…”

“…[15] The least common multiple of consecutive terms in a quadratic progression 403 P 4.4. Let k be any given positive integer.…”

Untitled

“…Then we see from [17] and [12,Theorem 4] that the left-hand side of (2.11) is divisible by a prime exceeding k. Furthermore, by [12, Theorem 4 0 ], the left-hand side of (2.11) is divisible by at least two distinct primes exceeding k whenever t ¼ k 5 4. Thus we see from (2.11), (2.6) and (2.7) that n þ ðk À 1Þd 5 q Analogously, we partition the set of a i 's in the following way.…”

On a generalization of a problem of Erdos and Graham

“…Obláth [25] obtained a similar statement for k = 5. Saradha and Shorey [30] proved that (1) has no solutions with k ≥ 4, provided that d is a power of a prime number. Laishram and Shorey [23] extended this result to the case where either d ≤ 10 10 , or d has at most six prime divisors.…”