Lagrange's four-square theorem

Lagrange's 4-square theorem states that every natural number can be represented as the sum of 4 integer squares.

The theorem was proven by Joseph Louis Lagrange in 1770.

Theorem statement

$$\displaystyle{ \forall n . n, S_i \in \mathbb{N} \to n = \sum_{i=1}^{4} S_i^2 }$$

Examples

$n = a^2 + b^2 + c^2 + d^2 \quad (n,a,b,c,d \in \mathbb{N})$

$$3 = 1^2 + 1^2 + 1^2 + 0^2 \\ 44 = 6^2 + 2^2 + 2^2 \\ 53 = 7^2 + 2^2$$

Blurb

squares:  1  2  3  4  5  6  7  8   9  10  11
n^2   = { 1, 4, 9,16,25,36,49,64, 81,100,121 s
n^2+1 = { 2, 5,10,17,26,37,50,65, 82,101,122 s+1
n^2+4 = { 5, 8,13,20,29,40,53,68, 85,104,125 s+4
n^2+9 = {10,13,18,25,34,45,58,73, 90,109,130 s+9
n^2+16= {17,20,25,32,41,52,65,80, 97,116,137 s+16
n^2+25= {26,29,34,41,50,61,74,89,106,125,145 s+25
n^2+36= {37,40,45,52,61,72,

0  = {0,0,0,0} = {0}
1  = {1,0,0,0} = {1}          1
2  = {1,1,0,0} = {1+1}        2
3  = {1,1,1,0} = {1+1+1}      3
4  = {1,1,1,1} = {1+1+1+1}    4

4  = {2,0,0,0} = {4}          1
5  = {2,1,0,0} = {4+1}        2
6  = {2,1,1,0} = {4+1+1}      3
7  = {2,1,1,1} = {4+1+1+1}    4 ++
8  = {2,2,0,0} = {4+4}        2
9  = {2,2,1,0} = {4+4+1}
10 = {2,2,1,1} = {4+4+1+1}
12 = {2,2,2,0} = {4+4+4}      3
13 = {2,2,2,1} = {4+4+4+1}
16 = {2,2,2,2} = {4+4+4+4}


9  = {3,0,0,0} = {9}          1
10 = {3,1,0,0} = {9+1}        2
11 = {3,1,1,0} = {9+1+1}      3
12 = {3,1,1,1} = {9+1+1+1}
13 = {3,2,0,0} = {9+4}        2
14 = {3,2,1,0} = {9+4+1}      3
15 = {3,2,1,1} = {9+4+1+1}    4
17 = {3,2,2,0} = {9+4+4}
18 = {3,2,2,1} = {9+4+4+1}
18 = {3,3,0,0} = {9+9}
19 = {3,3,1,0} = {9+9+1}
20 = {3,3,1,1} = {9+9+1+1}
22 = {3,3,2,0} = {9+9+4}
23 = {3,3,2,1} = {9+9+4+1}    4
26 = {3,3,2,2} = {9+9+4+4}
27 = {3,3,3,0} = {9+9+9}
28 = {3,3,3,1} = {9+9+9+1}    4
31 = {3,3,3,2} = {9+9+9+4}    4
36 = {3,3,3,3} = {9+9+9+9}
47 = {5,3,3,2} = {25+9+9+4}   4
39 = {6,1,1,1} = {36+1+1+1}   4
71 = {6,5,3,1} = {36+25+9+1}  4
55 = {7,2,1,1} = {49+4+1+1}   4
60 = {7,3,1,1} = {49+9+1+1}   4
63 = {7,3,2,1} = {49+9+4+1}   4

                           4^1               4^1               112
n=4^a(8b+7)={7 15 23 28 31 39 47 55 60 63 71|79 87 92 95 103 111 119 124 127
              +8 +8 +5 +3 +8 +8 +8 +5 +3 +8 +8 +8 +5 +3 +8  +8  +8  +5  +3


4^a x (8 x b + 7)
4^0 x (8 x 0 + 7) = 1 x (0 + 7) = 7
4^0 x (8 x 1 + 7) = 1 x (8 + 7) = 15
4^0 x (8 x 2 + 7) = 1 x (16+ 7) = 23
4^1 x (8 x 0 + 7) = 4 x (7)     = 28

8 x b + 7
8 x 0 + 7 = 7
8 x 1 + 7 = 15 = 8 x 2 - 1
8 x 2 + 7 = 23 = 8 x 3 - 1
8 x 3 + 7 = 31 = 8 x 4 - 1
8 x 4 + 7 = 39
8 x 5 + 7 = 47
8 x 6 + 7 = 55
8 x 7 + 7 = 63
8 x 8 + 7 = 71