Uniform lower bound for the least common multiple of a polynomial sequence
Shaofang Hong, Yuanyuan Luo, Guoyou Qian, Chunlin Wang · Comptes Rendus Mathématique · 2013
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 , except that f ( x ) = x and n = 1 , 2 , 3 , 4 , 6 and that f ( x ) = x s , with s ⩾ 2 being an integer and n = 1 , where ⌈ n / 2 ⌉ denotes the smallest integer, which is not less than n / 2 . This improves and extends the lower bounds obtained by M. Nair in 1982, B. Farhi in 2007 and S.M. Oon in 2013.