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ON THE NUMBER OF REPRESENTATIONS OF INTEGERS BY CERTAIN QUADRATIC FORMS

Part of: Forms and linear algebraic groups Other special functions Diophantine equations

Published online by Cambridge University Press:  01 August 2008

SHAUN COOPER
SHAUN COOPER*
Affiliation:
Institute of Information and Mathematical Sciences, Massey University, Private Bag 102904, North Shore Mail Centre, Auckland, New Zealand (email: s.cooper@massey.ac.nz)
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Abstract

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Generating functions are used to derive formulas for the number of representations of a positive integer by each of the quadratic forms x12+x22+x32+2x42, x12+2x22+2x32+2x42, x12+x22+2x32+4x42 and x12+2x22+4x32+4x42. The formulas show that the number of representations by each form is always positive. Some of the analogous results involving sums of triangular numbers are also given.

Keywords

sum of squaressum of triangular numbersquadratic formtheta functionelliptic functionLambert seriesLagrange’s four squares theorem

MSC classification

Secondary: 11E25: Sums of squares and representations by other particular quadratic forms 11D85: Representation problems 33E05: Elliptic functions and integrals
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 78 , Issue 1 , August 2008 , pp. 129 - 140
Copyright
Copyright © 2008 Australian Mathematical Society

References

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