# On the Sum of Consecutive Squares

• H. T. Freitag
• G. M. Phillips

## Abstract

We seek solutions of the Diophantine equation
$${\left( {n + 1} \right)^2} + {\left( {n + 2} \right)^2} + \cdots + {\left( {n + k} \right)^2} = {m^2}$$
(1)
for integers n, k and m, with n ≥ − 1 and with k and m positive and, in §3, we will also briefly consider the related equation where m 2 is replaced by m 3. When k = 1, (1) is trivial. For k = 2 we replace n by n − 1 and express (1) in the form
$${n^2} + {\left( {n + 1} \right)^2} = {m^2}.$$
(2)

## Keywords

Positive Integer Related Equation Number Theory Difference Equation Infinite Number
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## References

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Philipp, S. “Note on consecutive integers whose sum of squares is a perfect square”. Mathematics Magazine, Vol. 37(1964): pp. 218–220.