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ON THE NUMBER OF REPRESENTATIONS OF INTEGERS BY CERTAIN QUADRATIC FORMS

Published online by Cambridge University Press:  01 August 2008

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.

Type
Research Article
Copyright
Copyright © 2008 Australian Mathematical Society

References

[1]Adiga, C., Cooper, S. and Han, J. H., ‘A general relation between sums of squares and sums of triangular numbers’, Int. J. Number Theory 1 (2005), 175182.CrossRefGoogle Scholar
[2]Baruah, N. D., Cooper, S. and Hirschhorn, M., ‘Sums of squares and sums of triangular numbers induced by partitions of 8’, Int. J. Number Theory to appear.Google Scholar
[3]Berndt, B. C., Ramanujan’s Notebooks, Part III (Springer-Verlag, New York, 1991).CrossRefGoogle Scholar
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ON THE NUMBER OF REPRESENTATIONS OF INTEGERS BY CERTAIN QUADRATIC FORMS
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