Uniform lower bound for the least common multiple of a polynomial sequence

Let n   be a positive integer and f(x)f(x) be a polynomial with nonnegative integer coefficients. We prove that lcm⌈n/2⌉⩽i⩽n{f(i)}⩾2nlcm⌈n/2⌉⩽i⩽n{f(i)}⩾2n, except that f(x)=xf(x)=x and n=1,2,3,4,6n=1,2,3,4,6 and that f(x)=xsf(x)=xs, with s⩾2s⩾2 being an integer and n=1n=1, where ⌈n/2⌉⌈n/2⌉ denotes the smallest integer, which is not less than n/2n/2. This improves and extends the lower bounds obtained by M. Nair in 1982, B. Farhi in 2007 and S.M. Oon in 2013.

RésuméSoit n   un entier ⩾1 et f(x)f(x) un polynôme à coefficients entiers ⩾0. Nous démontrons que, à lʼexception de certains cas explicites, on a ppcm⌈n/2⌉⩽i⩽n{f(i)}⩾2nppcm⌈n/2⌉⩽i⩽n{f(i)}⩾2n, où ⌈n/2⌉⌈n/2⌉ dénote le plus petit entier ⩾n/2⩾n/2. Ceci améliore, et étend, les bornes inférieures obtenues par M. Nair en 1982, B. Farhi en 2007 et S.M. Oon en 2013.