Further Improvements of Lower Bounds for the Least Common Multiples of Arithmetic Progressions

2008/11/28 by Shaofang Hong, Hong, Shaofang, Scott D. Kominers +1
Mathematics · #11A05 #FOS: Mathematics #Number Theory (math.NT) #math.NT #msc:11A05

paper · pdf · doi:10.48550/arxiv.0811.4769

5 pages; v3: improved results and updated references

arxiv created 2009/06/16 · arxiv updated 2009/12/01

Abstract

For relatively prime positive integers u₀ and r, we consider the arithmetic progression u_k := u₀+k*r (0 <= k <= n). Define L_n := lcmu₀,u₁,...,u_n and let a >= 2 be any integer. In this paper, we show that, for integers alpha,r >= a and n >= 2*alpha*r, we have L_n >= u₀*r^alpha+a-2*(r+1)ⁿ. In particular, letting a = 2 yields an improvement to the best previous lower bound on L_n (obtained by Hong and Yang) for all but three choices of alpha,r >= 2.

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