Nontrivial lower bounds for the least common multiple of some finite sequences of integers

Abstract

We present here a method which allows to derive a nontrivial lower bounds for the least common multiple of some finite sequences of integers. We obtain efficient lower bounds (which in a way are optimal) for the arithmetic progressions and lower bounds less efficient (but nontrivial) for quadratic sequences whose general term has the form u n = an(n + t) + b with (a, t, b) ∈ Z 3 , a 5, t 0, gcd(a, b) = 1. From this, we deduce for instance the lower bound: lcm{1 2 + 1, 2 2 + 1, . . . , n 2 + 1} 0, 32(1, 442) n (for all n 1). In the last part of this article, we study the integer lcm(n, n + 1, . . . , n + k) (k ∈ N, n ∈ N * ). We show that it has a divisor d n,k simple in its dependence on n and k, and a multiple m n,k also simple in its dependence on n. In addition, we prove that both equalities: lcm(n, n + 1, . . . , n+ k) = d n,k and lcm(n, n + 1, . . . , n+ k) = m n,k hold for an infinitely many pairs (n, k).

The Least Common Multiple of Consecutive Terms in a Quadratic Progression

“…to lcm 1≤i≤k {i} by showing thatḡ k (1) |ḡ k (n) for any positive integer n. Moreover, they conjectured that lcm 1≤i≤k+1 {i}/(k + 1) dividesP k for all nonnegative integers k. Farhi and Kane [5] confirmed the Hong-Yang conjecture and determined the exact value ofP k . Note that Farhi [4] also obtained the following nontrivial lower bound: lcm 1≤i≤n {i 2 + 1} ≥ 0.32 · (1.442) n (for all n ≥ 1).…”

“…In [4], Farhi investigated the least common multiple lcm 0≤i≤k {n + i} of finitely many consecutive integers by introducing the arithmetic function g k (n) := k i=0 (n + i) lcm 0≤i≤k {n + i} , and also proved some arithmetic properties of lcm 0≤i≤k {n + i}. Farhi showed thatḡ k is periodic and k!…”

“…ThenP k | k!. But Farhi did not determine the exact value ofP k in [4], so he posed the open problem of determining the smallest periodP k . Hong and Yang [11] improved the period k!…”

Integral Difference Ratio Functions on Integers

“…Definition 11) has many interesting properties and recently regained interest, cf. [12,9,5,6,4]. We state four lemmas used in the proof of Theorem 16.…”

Improvements of lower bounds for the least common multiple of finite arithmetic progressions

“…Hanson [6] and Nair [12] derived the upper bound and lower bound of lcm{1, · · · , n} respectively. Farhi [3], [4] obtained non-trivial lower bounds for the least common multiple of some finite arithmetic progressions. In what follows we always let u 0 , r and n be positive integers such that u 0 and r are coprime.…”