New lower bounds for the least common multiples of arithmetic progressions

Abstract

For relatively prime positive integers u ₀ and r, and for 0 ≤ k ≤ n, define u _k:= u ₀ + kr. Let L _n:= lcm(u ₀, u ₁, ..., u _n) and let a, l ≥ 2 be any integers. In this paper, the authors show that, for integers α ≥ a, r ≥ max(a,l − 1) and n ≥ lαr, the following inequality holds $L_n \geqslant u_0 r^{\left( {l - 1} \right)\alpha + a - l} \left( {r + 1} \right)^n .$ Particularly, letting l = 3 yields an improvement on the best previous lower bound on L _n obtained by Hong and Kominers in 2010.

Keywords

Arithmetic progression / Least common multiple / Lower bound

References

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