Sums of both 2 and 3 Consecutive Squares

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

1988: Martin Gardner: Time Travel and Other Mathematical Bewilderments: $2$: Hexes and Stars
1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $365$