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| New Lower Bounds for the Least Common Multiples of Arithmetic Progressions |
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Citation： |
Rongjun WU,Qianrong TAN,Shaofang HONG.New Lower Bounds for the Least Common Multiples of Arithmetic Progressions[J].Chinese Annals of Mathematics B,2013,34(6):861~864 |
| Page view： 1614
Net amount： 1178 |
Authors： |
Rongjun WU; Qianrong TAN; Shaofang HONG; |
Foundation： |
National Natural Science Foundation of China (No. 10971145), the Ph.D.Programs Foundation of Ministry of Education of China (No. 20100181110073) and the Science & TechnologyProgram of Sichuan Province (No. 2013JY0125). |
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| Abstract： |
For relatively prime positive integers u0 and r, and for 0 ≤ k ≤ n, define
uk := u0 + kr. Let Ln := lcm(u0, u1, · · · , un) 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
Ln ≥ u0r(l?1)α+a?l(r +1)n.
Particularly, letting l = 3 yields an improvement on the best previous lower bound on Ln
obtained by Hong and Kominers in 2010. |
Keywords： |
Arithmetic progression, Least common multiple, Lower bound |
Classification： |
11B25, 11N13, 11A05 |
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