New lower bounds for the least common multiple of polynomial sequences
New lower bounds for the least common multiple of polynomial sequences
复制标题
多项式序列最小公倍数的新下界
DOI：
10.1016/j.jnt.2016.11.026
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发表时间：
2017-06
影响因子：
0.7
通讯作者：
Qian Guoyou
中科院分区：
文献类型：
--
作者：
Hong Shaofang;Qian Guoyou
设n是正整数，f(X)是具有非负整系数的多项式。证明了对任意正整数n，lcm⌈n/2⌉≤i≤n{f(I)}≥2 n−1⌈n/2⌉，其中⌈n/2⌉表示不小于n/2的最小整数，改进了Hong，Luo，Qian和Wang在2013年得到的下界.对于前n个正整数中的最小公倍数，我们证明了LCM1≤i≤n{i}≥2 n−3(n−1)n−2 2是任意整数n≥7的下界，改进了Nair在1982年和Farhi在2009年得到的下界。对于连续二次级数的最小公倍数，利用积分的方法，结合对复数绝对值的更详细的分析，进一步证明了对任意正整数n，a和c，lcm⌈n/2⌉≤i≤n{a I2+c}≥2 n−1⌈n/2⌉⋅min(a，a c).这改进和扩展了Farhi在2005年和Oon在2013年分别获得的结果。
Let n be a positive integer and f (x) be a polynomial with nonnegative integer coefficients. We prove that lcm⌈ n/2⌉≤ i≤ n {f (i)}≥ 2 n− 1⌈ n/2⌉ for any positive integer n, where⌈ n/2⌉ denotes the smallest integer that is not less than n/2. This improves the lower bound obtained by Hong, Luo, Qian and Wang in 2013. For the least common multiple of the first n positive integers, we show that lcm 1≤ i≤ n {i}≥ 2 n− 3 (n− 1) n− 2 2 for any integer n≥ 7, which improves the lower bound obtained by Nair in 1982 and by Farhi in 2009. For the least common multiple of consecutive quadratic progression terms, by using the integration method combined with a little more detailed analysis on the absolute value of complex numbers, we further show that lcm⌈ n/2⌉≤ i≤ n {a i 2+ c}≥ 2 n− 1⌈ n/2⌉⋅ min(a, a c) for any positive integer n, where a and c are two positive integers. This improves and extends the results obtained by Farhi in 2005 and Oon in 2013, respectively.
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DOI：
10.1090/s0002-9939-08-09730-x
发表时间：
2008-08
期刊：
--
影响因子：
--
作者：
Bakir Farhi;D. Kane
通讯作者：
Bakir Farhi;D. Kane
影响因子：
0.8
作者：
Bakir Farhi
通讯作者：
Bakir Farhi
影响因子：
0.6
作者：
Guoyou Qian;Shaofang Hong
通讯作者：
Guoyou Qian;Shaofang Hong
影响因子：
5
作者：
Edmund Taylor Whittaker;G. Watson
通讯作者：
Edmund Taylor Whittaker;G. Watson
影响因子：
0.9
作者：
Shaofang Hong;Guoyou Qian;Qianrong Tan
通讯作者：
Qianrong Tan