Sums of both 2 and 3 Consecutive Squares

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Theorem

The following are the smallest positive integers that are the sum of both $2$ and $3$ consecutive non-zero square numbers:

$365, 35 \, 645, 3 \, 492 \, 725, 342 \, 251 \, 285, 33 \, 537 \, 133 \, 085, 3 \, 286 \, 296 \, 790 \, 925, \ldots$

This sequence is A007667 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).


Proof

\(\ds 365\) \(=\) \(\ds 10^2 + 11^2 + 12^2\)
\(\ds \) \(=\) \(\ds 13^2 + 14^2\)


\(\ds 35 \, 645\) \(=\) \(\ds 108^2 + 109^2 + 110^2\)
\(\ds \) \(=\) \(\ds 133^2 + 134^2\)


\(\ds 3 \, 492 \, 725\) \(=\) \(\ds 1078^2 + 1079^2 + 1080^2\)
\(\ds \) \(=\) \(\ds 1321^2 + 1322^2\)


\(\ds 342 \, 251 \, 285\) \(=\) \(\ds 10 \, 680^2 + 10 \, 681^2 + 10 \, 682^2\)
\(\ds \) \(=\) \(\ds 13 \, 081^2 + 13 \, 082^2\)


\(\ds 33 \, 537 \, 133 \, 085\) \(=\) \(\ds 105 \, 730^2 + 105 \, 731^2 + 105 \, 732^2\)
\(\ds \) \(=\) \(\ds 129 \, 493^2 + 129 \, 494^2\)


\(\ds 3 \, 286 \, 296 \, 790 \, 925\) \(=\) \(\ds 1 \, 046 \, 628^2 + 1 \, 046 \, 629^2 + 1 \, 046 \, 630^2\)
\(\ds \) \(=\) \(\ds 1 \, 281 \, 853^2 + 1 \, 281 \, 854^2\)


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Sources

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