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    This article has been cited by the following publications. This list is generated based on data provided by CrossRef.

    Hakami, A. H. 2015. Small primitive zeros of quadratic forms mod $$p^m$$ p m. The Ramanujan Journal, Vol. 38, Issue. 1, p. 189.

    Heath-Brown, D. R. 1991. Small solutions of quadratic congruences, II. Mathematika, Vol. 38, Issue. 02, p. 264.


Small zeros of quadratic congruences modulo pq

  • Todd Cochrane (a1)
  • DOI:
  • Published online: 01 February 2010

Let Q(x) = Q(x1, x2,…, xn) be a quadratic form with integer coefficients. Schinzel, Schickewei and Schmidt [9, Theorem 1] have shown that for any modulus m there exists a nonzero such that

and ║x║≤m(1/2)+(1/2(n-1)), where ║x║ = max |xi|. When m is a prime Heath-Brown [8] has obtained a nonzero solution of (1) with ║x║≤m1/2 log m. Yuan [10] has extended Heath-Brown's work to all finite fields. We have proved related results in [5] and [6]. In this paper we extend Heath-Brown's work to moduli which are a product of two primes. Throughout the paper we shall assume that n is even and n>2. For any odd prime p let

where det Q is the determinant of the integer matrix representing Q and is the Legendre symbol.

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2.L. Carlitz . Weighted quadratic partitions over a finite field. Can. J. of Math., 5 (1953), 317323.

5.T. Cochrane . Small zeros of quadratic forms modulo p. J. of Number Theory, 33 (1989), 286292.

10.W. Yuan . On small zeros of quadratic forms over finite fields. J. of Number Theory, 31 (1989), 272284.

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  • ISSN: 0025-5793
  • EISSN: 2041-7942
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