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Sums of squares of integers in arithmetic progression

Published online by Cambridge University Press:  23 January 2015

Tom M. Apostol  and
Mamikon A. Mnatsakanian
Tom M. Apostol
Affiliation:
Project MATHEMATICS!, California Institute of Technology, Pasadena, CA 91125 USA
Mamikon A. Mnatsakanian
Affiliation:
Project MATHEMATICS!, California Institute of Technology, Pasadena, CA 91125 USA
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Extract

The following striking identities

are the cases n = 1,2,3,4 of a remarkable family given by G.J. Dostor [1]:

where m = n(2n + 1), and n = 1, 2, … The case m = −n is trivial. If m ≠ −n there are n + 1 squares of consecutive integers on the left and n on the right. We will treat the last term (m + n)2 on the left differently, and refer to it as a transition term relating two sums of squares of n consecutive integers.

Type
Articles
Information
The Mathematical Gazette , Volume 95 , Issue 533 , July 2011 , pp. 186 - 196
Copyright
Copyright © The Mathematical Association 2012

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References

1. Dostor, G. J., Questions sur les nombres, Archiv der Mathematik und Physik, 64 (1879), pp. 350–352.Google Scholar
2. Dickson, L. E., A history of the theory of numbers, Vol. II (Carnegie Institute, Washington, DC, 1920; reprinted Chelsea Pub., New York, NY, 1952; Dover Pub. Inc., New York, NY, 2005).Google Scholar