Integer as Sum of 5 Non-Zero Squares

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Theorem

Let $n \in \Z$ be an integer such that $n > 33$.

Then $n$ can be expressed as the sum of $5$ non-zero squares.


Proof

From Lagrange's Four Square Theorem, every positive integer can be expressed as the sum of $4$ squares, some of which may be zero.

The existence of positive integers which cannot be expressed as the sum of $4$ non-zero squares is noted by the trivial examples $1$, $2$ and $3$.

Thus Lagrange's Four Square Theorem can be expressed in the form:

$(1): \quad$ Every positive integer can be expressed as the sum of $1$, $2$, $3$ or $4$ non-zero squares.


We note the following from 169 as Sum of up to 155 Squares:

\(\displaystyle 169\) \(=\) \(\displaystyle 13^2\)
\(\displaystyle \) \(=\) \(\displaystyle 12^2 + 5^2\)
\(\displaystyle \) \(=\) \(\displaystyle 12^2 + 4^2 + 3^2\)
\(\displaystyle \) \(=\) \(\displaystyle 8^2 + 8^2 + 5^2 + 4^2\)
\(\displaystyle \) \(=\) \(\displaystyle 8^2 + 8^2 + 4^2 + 4^2 + 3^2\)

Let $n > 169$.

Then $n$ can be expressed as:

$n = m + 169$

where $m \ge 1$.

From $(1)$, $m$ can be expressed as the sum of sum of $1$, $2$, $3$ or $4$ non-zero squares.

Thus at least one of the following holds:

$m = a^2$
$m = a^2 + b^2$
$m = a^2 + b^2 + c^2$
$m = a^2 + b^2 + c^2 + d^2$


Thus one of the following holds:

\(\displaystyle n\) \(=\) \(\displaystyle a^2 + b^2 + c^2 + d^2 + 13^2\)
\(\displaystyle n\) \(=\) \(\displaystyle a^2 + b^2 + c^2+ 12^2 + 5^2\)
\(\displaystyle n\) \(=\) \(\displaystyle a^2 + b^2 + 12^2 + 4^2 + 3^2\)
\(\displaystyle n\) \(=\) \(\displaystyle a^2 + 8^2 + 8^2 + 5^2 + 4^2\)


It remains to be shown that of the positive integers less than $169$, all but the following can be expressed in this way:

$1, 2, 3, 4, 6, 7, 9, 10, 12, 15, 18, 33$


First we show the ones which can:

\(\displaystyle 5\) \(=\) \(\displaystyle 1^2 + 1^2 + 1^2 + 1^2 + 1^2\)
\(\displaystyle 8\) \(=\) \(\displaystyle 2^2 + 1^2 + 1^2 + 1^2 + 1^2\)
\(\displaystyle 11\) \(=\) \(\displaystyle 2^2 + 2^2 + 1^2 + 1^2 + 1^2\)
\(\displaystyle 13\) \(=\) \(\displaystyle 3^2 + 1^2 + 1^2 + 1^2 + 1^2\)
\(\displaystyle 14\) \(=\) \(\displaystyle 2^2 + 2^2 + 2^2 + 1^2 + 1^2\)
\(\displaystyle 16\) \(=\) \(\displaystyle 3^2 + 2^2 + 1^2 + 1^2 + 1^2\)
\(\displaystyle 17\) \(=\) \(\displaystyle 2^2 + 2^2 + 2^2 + 2^2 + 1^2\)
\(\displaystyle 19\) \(=\) \(\displaystyle 3^2 + 2^2 + 2^2 + 1^2 + 1^2\)
\(\displaystyle 20\) \(=\) \(\displaystyle 2^2 + 2^2 + 2^2 + 2^2 + 2^2\)
\(\displaystyle 21\) \(=\) \(\displaystyle 3^2 + 3^2 + 1^2 + 1^2 + 1^2\)
\(\displaystyle 22\) \(=\) \(\displaystyle 3^2 + 2^2 + 2^2 + 2^2 + 1^2\)
\(\displaystyle 23\) \(=\) \(\displaystyle 4^2 + 2^2 + 1^2 + 1^2 + 1^2\)
\(\displaystyle 24\) \(=\) \(\displaystyle 3^2 + 3^2 + 2^2 + 1^2 + 1^2\)
\(\displaystyle 25\) \(=\) \(\displaystyle 3^2 + 2^2 + 2^2 + 2^2 + 2^2\)
\(\displaystyle 26\) \(=\) \(\displaystyle 4^2 + 2^2 + 2^2 + 1^2 + 1^2\)
\(\displaystyle 27\) \(=\) \(\displaystyle 3^2 + 3^2 + 2^2 + 2^2 + 1^2\)
\(\displaystyle 28\) \(=\) \(\displaystyle 4^2 + 3^2 + 1^2 + 1^2 + 1^2\)
\(\displaystyle 29\) \(=\) \(\displaystyle 3^2 + 3^2 + 3^2 + 1^2 + 1^2\)
\(\displaystyle 30\) \(=\) \(\displaystyle 3^2 + 3^2 + 2^2 + 2^2 + 2^2\)
\(\displaystyle 31\) \(=\) \(\displaystyle 4^2 + 3^2 + 2^2 + 1^2 + 1^2\)
\(\displaystyle 32\) \(=\) \(\displaystyle 4^2 + 2^2 + 2^2 + 2^2 + 2^2\)
\(\displaystyle 34\) \(=\) \(\displaystyle 4^2 + 3^2 + 2^2 + 2^2 + 1^2\)
\(\displaystyle 35\) \(=\) \(\displaystyle 4^2 + 4^2 + 1^2 + 1^2 + 1^2\)
\(\displaystyle 36\) \(=\) \(\displaystyle 4^2 + 3^2 + 3^2 + 1^2 + 1^2\)
\(\displaystyle 37\) \(=\) \(\displaystyle 3^2 + 3^2 + 3^2 + 3^2 + 1^2\)
\(\displaystyle 38\) \(=\) \(\displaystyle 4^2 + 4^2 + 2^2 + 1^2 + 1^2\)
\(\displaystyle 39\) \(=\) \(\displaystyle 4^2 + 3^2 + 3^2 + 2^2 + 1^2\)
\(\displaystyle 40\) \(=\) \(\displaystyle 3^2 + 3^2 + 3^2 + 3^2 + 2^2\)
\(\displaystyle 41\) \(=\) \(\displaystyle 4^2 + 4^2 + 2^2 + 2^2 + 1^2\)
\(\displaystyle 42\) \(=\) \(\displaystyle 4^2 + 3^2 + 3^2 + 2^2 + 2^2\)



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