New lower bounds for the least common multiple of polynomial sequences

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)}≥2n−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 lcm1≤i≤n{i}≥2n−3(n−1)n−22 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{ai2+c}≥2n−1⌈n/2⌉⋅min⁡(a,ac) 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.