Numbers Expressible as Sum of Five Distinct Squares

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Theorem

The largest positive integer which cannot be expressed as the sum of no more than $5$ distinct squares is $188$.


Both $188$ and $124$ require as many as $6$ distinct squares to represent them:

\(\ds 124\) \(=\) \(\ds 1 + 4 + 9 + 25 + 36 + 49\)
\(\ds \) \(=\) \(\ds 1^2 + 2^2 + 3^2 + 5^2 + 6^2 + 7^2\)
\(\ds 188\) \(=\) \(\ds 1 + 4 + 9 + 25 + 49 + 100\)
\(\ds \) \(=\) \(\ds 1^2 + 2^2 + 3^2 + 5^2 + 7^2 + 10^2\)


Proof

From Numbers not Sum of Distinct Squares, the following positive integers cannot be expressed as the sum of distinct squares at all:

$2, 3, 6, 7, 8, 11, 12, 15, 18, 19, 22, 23, 24, 27, 28, 31, 32, 33, 43, 44, 47, 48, 60, 67, 72, 76, 92, 96, 108, 112, 128$

This sequence is A001422 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).



This theorem requires a proof.
In particular: It remains to be shown that all other positive integers less than $188$ can be expressed as the sum of no more than 5 distinct squares, and all numbers over $188$ can also be so expressed.
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Sources

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