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

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

2008-11-28

arxiv.org/abs/0811.4769v3

Mathematics

Number Theory

5 pages; v3: improved results and updated references

Scientific paper

For relatively prime positive integers u_0 and r, we consider the arithmetic progression {u_k := u_0+k*r} (0 <= k <= n). Define L_n := lcm{u_0,u_1,...,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 >= u_0*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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