Small zeros of quadratic congruences modulo pq

Primitive zeros of quadratic forms mod $p^2$

“…Thus for any ε > 0 we get a nonzero solution of (1) For m = pq a product of two distinct primes, the optimal bound, ∥x∥ ≪ m 1/2 for n > 4 was obtained by Cochrane [4] and [5], building upon the work of Heath-Brown [14]. But no attempt was made to obtain a primitive solution in this work.…”

“…In the case where l = 0, the theorem gives us a primitive solution of (2) with ∥x∥ ≤ 2 4+ 3 n p, recovering the type of bound obtained in [8] and [6]. To prove these results we shall use finite Fourier series over Z p 2 , the modular ring in p 2 elements.…”

“…To prove these results we shall use finite Fourier series over Z p 2 , the modular ring in p 2 elements. The proof here builds upon the work of [6] and [8].…”

Small zeros of quadratic forms mod P^m

“…Wang Yuan [16], [17] and [18] generalized Heath-Brown's work to all finite fields. Cochrane, in a sequence of papers [1], [2] and [3] [5] sharpened this result to x M 1/2 , for n > 4. Our purpose in this paper is to generalize (modp) methods for obtaining a small primitive solution of…”

Small solutions of quadratic congruences, II

“…In the case in which q is prime and all the K t are equal, the author [8] succeeded in showing that B n (q) < q i/2 logq for H3=4, using methods quite different from those Schinzel, Schlickewei and Schmidt [10], and Baker and Harman [2]. Very recently Cochrane [5] sharpened the argument slightly to show that B n (q) < q i/2 for n 3=4, again for q prime and the K, all equal. As we have seen, this bound is best possible.…”