Legendre's three-square theorem

In mathematics, Legendre's three-square theorem states that a natural number can be represented as the sum of three squares of integers

n=x^{2}+y^{2}+z^{2}

if and only if $n$ is not of the form $n=4^{a}(8b+7)$ for nonnegative integers $a$ and $b$ .

The first numbers that cannot be expressed as the sum of three squares (i.e. numbers that can be expressed as $n=4^{a}(8b+7)$ ) are

7, 15, 23, 28, 31, 39, 47, 55, 60, 63, 71 ... (sequence A004215 in the OEIS).

a b	0	1	2
0	7	28	112
1	15	60	240
2	23	92	368
3	31	124	496
4	39	156	624
5	47	188	752
6	55	220	880
7	63	252	1008
8	71	284	1136
9	79	316	1264
10	87	348	1392
11	95	380	1520
12	103	412	1648
Unexpressible values up to 100 are in bold

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