A little while ago I noticed a pattern in the sums of the digits of perfect squares that seems to suggest that:(adsbygoogle = window.adsbygoogle || []).push({});

For a natural number N, the digits of N^2 add up to either 1, 4, 7, or 9.

ex: 5^2 = 25, 2+5 = 7

In some cases, the summation must be iterated several times:

ex: 7^2 = 49, 4+9=13, 1+3 = 4

ex: 10^2 = 100, 1+0+0 = 1

If you make a list of the summations of the perfect square digits, the following patterns emerge:

Sums of digits (of the following numbers ^ 2) -

1 > 1, 8, 10, 17, 19...

4 > 2, 7, 11, 16, 20...

7 > 4, 5, 13, 14, 22, 23...

9 > 3, 6, 9, 12, 15...

(As an extension, this pattern seems to hold up for N ^ any even power...this might help in formulating a proof. )

Does anyone know if there exists a proof for this apparent pattern, or does anyone have any idea of how to formulate a proof for it?

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# Sums of digits of Perfect Squares

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