On pairings of the first 2n natural numbers
Let denote the number of partitions of into parts, each of which is at least . By applying the saddle point method to the generating series, an asymptotic estimate is given for , which holds for , and .
It is proved that, if F(x) be a cubic polynomial with integral coefficients having the property that F(n) is equal to a sum of two positive integral cubes for all sufficiently large integers n, then F(x) is identically the sum of two cubes of linear polynomials with integer coefficients that are positive for sufficiently large x. A similar result is proved in the case where F(n) is merely assumed to be a sum of two integral cubes of either sign. It is deduced that analogous propositions are true...