Longer Gaps Between Values of Binary Quadratic Forms

We prove new lower bounds on large gaps between integers that are sums of two squares or are represented by any binary quadratic form of discriminant D, improving the results of Richards. Let s₁, s₂, . . . be the sequence of positive integers, arranged in increasing order, that are representable by any binary quadratic form of fixed discriminant D, then (Formula presented) improving a lower bound of 1/ |D | of Richards. In the special case of sums of two squares, we improve Richards’s bound of 1/4 to 390/449 = 0.868 . . .. We also generalize Richards’s result in another direction: if d is composite we show that there exist constants C_d such that for all integer values of x none of the values p_d(x) = C_d+x^d is a sum of two squares. Let d be a prime. For all k ∈ N, there exists a smallest positive integer y_k such that none of the integers y_k + j^d, 1 ≤ j ≤ k, is a sum of two squares.