Lower bounds for the least common multiple of finite arithmetic progressions

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

“…Note that this conjecture extends Nair's result [12] from the set {1, · · · , n} to a general arithmetic progression with n terms. Hong and Feng [8] confirmed Farhi's conjecture. Meanwhile, Hong and Feng [8] obtained an improved lower bound under certain conditions.…”

“…Hong and Feng [8] confirmed Farhi's conjecture. Meanwhile, Hong and Feng [8] obtained an improved lower bound under certain conditions. In fact, they proved that L n ≥ u 0 r(r + 1) n if n > r. In this paper, our main interest is the lower bounds for the least common multiple of finite arithmetic progressions.…”

“…By [8], we know that Theorem 1.1 is true if α = 1 or r = 1. It remains to consider the case where α ≥ 2 and r ≥ 2.…”

New results on the least common multiple of consecutive integers

“…, u n ) in the cases where (u n ) n is an arithmetic progression and where it is a quadratic progression. In the case of arithmetic progressions, Hong and Feng [7] and Hong and Yang [8] obtained some improvements of Farhi's lower bounds.…”

Divisibility properties of Smith matrices

“…The renowned Dirichlet theorem states that the arithmetic progression contains infinitely many primes if the first term and the common difference are coprime, while the Green-Tao theorem [11] says that the set of primes contains arbitrarily long arithmetic progressions. Farhi [9] and Hong-Feng [20] investigated the non-trivial lower bounds for the least common multiple of finite arithmetic progressions. Ligh [25] raised the problem of computing the determinants of Smith matrices on a finite arithmetic progression which is still open.…”