Publications

Publications

- March 2010
- Proceedings of the American Mathematical Society

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

By: Shaofang Hong and Scott Duke Kominers

## Abstract

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.

## Keywords

## Citation

Hong, Shaofang, and Scott Duke Kominers. "Further Improvements of Lower Bounds for the Least Common Multiples of Arithmetic Progressions."

*Proceedings of the American Mathematical Society*138, no. 3 (March 2010): 809–813.