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ERDŐS–TURÁN WITH A MOVING TARGET, EQUIDISTRIBUTION OF ROOTS OF REDUCIBLE QUADRATICS, AND DIOPHANTINE QUADRUPLES

Part of: Diophantine equations Probabilistic theory: distribution modulo $1$; metric theory of algorithms Multiplicative number theory

Published online by Cambridge University Press:  13 December 2010

Greg Martin  and
Scott Sitar
Greg Martin
Affiliation:
Department of Mathematics, University of British Columbia, Room 121, 1984 Mathematics Road, Canada V6T 1Z2 (email: [email protected])
Scott Sitar
Affiliation:
Department of Mathematics, University of British Columbia, Room 121, 1984 Mathematics Road, Canada V6T 1Z2 (email: [email protected])
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Abstract

A Diophantine m-tuple is a set A of m positive integers such that ab+1 is a perfect square for every pair a,b of distinct elements of A. We derive an asymptotic formula for the number of Diophantine quadruples whose elements are bounded by x. In doing so, we extend two existing tools in ways that might be of independent interest. The Erdős–Turán inequality bounds the discrepancy between the number of elements of a sequence that lie in a particular interval modulo 1 and the expected number; we establish a version of this inequality where the interval is allowed to vary. We also adapt an argument of Hooley on the equidistribution of solutions of polynomial congruences to handle reducible quadratic polynomials.

MSC classification

Secondary: 11D45: Counting solutions of Diophantine equations 11N45: Asymptotic results on counting functions for algebraic and topological structures 11K06: General theory of distribution modulo $1$ 11K38: Irregularities of distribution, discrepancy
Type
Research Article
Information
Mathematika , Volume 57 , Issue 1 , January 2011 , pp. 1 - 29
Copyright
Copyright © University College London 2011

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References

[1]Abramowitz, M. and Stegun, I. A., Handbook of Mathematical Functions, Dover Publications Inc. (New York, 1965).Google Scholar
[2]Dujella, A., There are only finitely many Diophantine quintuples. J. Reine Angew. Math. 566 (2004), 183–214.Google Scholar
[3]Dujella, A., On the number of Diophantine m-tuples. Ramanujan J. 15(1) (2008), 37–46.Google Scholar
[4]Hooley, C., On the number of divisors of quadratic congruences. Acta Math. 110 (1963), 97–114.Google Scholar
[5]Hooley, C., On the distribution of the roots of polynomial congruences. Mathematika 11 (1964), 39–49.Google Scholar
[6]Martin, G., An asymptotic formula for the number of smooth values of a polynomial. J. Number Theory 93(2) (2002), 108–182.CrossRefGoogle Scholar
[7]Montgomery, H. L., Ten Lectures on the Interface Between Analytic Number Theory and Harmonic Analysis (CBMS Regional Conference Series in Mathematics 84), American Mathematical Society (Providence, RI, 1994).CrossRefGoogle Scholar
[8]Niven, I., Zuckerman, H. S. and Montgomery, H. L., An Introduction to the Theory of Numbers, 5th edn, John Wiley & Sons, Inc. (New York, 1991).Google Scholar