Sums of squares

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All positive integers can be expressed as sums of squares. Some can be expressed as the sum of two or three squares, some can be expressed as the sum of a million squares. And some can be expressed expressed as sums of squares in multiple ways. For example, 338350 is the sum of the first hundred nonzero squares. It can also be represented as 580² + 43² + 10² + 1².

Since the square of a negative number is a positive number, we will not bother to distinguish between the squares of negative integers and the squares of positive integers in this article. However, we will distinguish between sums of squares that may include instances of 0² and those that must consist solely of the squares of nonzero integers; the latter will be referred to as "nonzero squares" here.

Among numbers that can be represented as the sum of a given number of squares, we may distinguish between those that can be represented with fewer squares and those that can't. For example, 25 = 4² + 3², but also 5², whereas 29 = 5² + 2² but we can't use $({\sqrt {29}})^{2}$ (that is inelegant at best and cheating at worst).

$k$	Numbers that can be represented with $k$ squares		Numbers that can't be represented with fewer than $k$ squares
1	1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, ...	A000290
2	2, 5, 8, 10, 13, 17, 18, 20, 25, 26, 29, 32, ...	A000404	2, 5, 8, 10, 13, 17, 18, 20, 26, 29, 32, 34, ...	A000415
3	3, 6, 9, 11, 12, 14, 17, 18, 19, 21, 24, 26, 27, 30, ...	A000408	3, 6, 11, 12, 14, 19, 21, 22, 24, 27, 30, 33, 35, 38, ...	A000419
4	4, 7, 10, 12, 13, 15, 16, 18, 19, 20, 21, 22, 23, 25, ...	A000414	7, 15, 23, 31, 39, 47, 55, 63, 71, 79, 87, 95, 103, ...	A004771
5	5, 8, 11, 13, 14, 16, 17, 19, 20, 21, 22, 23, 24, 25, ...	A047700	None
6	6, 9, 12, 14, 15, 16, 17, 20, 21, 22, ...
7	7, 10, 13, 15, 16, 18, 21, 22, ...

Sums of four squares

The answer to Waring's problem for squares is that all integers have at least one representation as the sum of at most four squares.

Theorem SQS4. Every positive integer can be expressed as the sum of at most four nonzero squares. Or we can say that all positive integers are the sum of four squares, some, but not all, of which may be zero: that is, $n=w^{2}+x^{2}+y^{2}+z^{2}$, with $w>x\geq y\geq z\geq 0$.

Proof. PROOF GOES HERE. ENDOFPROOFMARK

When $n\equiv 7\mod 8$, the number requires four nonzero squares. Obviously prime numbers require at least two nonzero squares for their Waring representation. See Theorem P2SQ in the Gaussian integers article for a result regarding which primes can be expressed as the sum of two squares.

Corollary to Theorem SQS4. If $n$ is the sum of four equal nonzero squares, then it has at least two representations as sum of squares, at least one of which includes zeroes. The stricture $w>x\geq y\geq z\geq 0$ may have given the impression that the theorem does not cover the sums of equal squares, but this would be a false impression, as a little rewriting will show. If $w=x=y=z>0$, then $n=4w^{2}$ (or $4x^{2}$, etc.). We can rewrite $n=4w^{2}=2^{2}w^{2}=(2w)^{2}$. Reassigning $w:=2w$ and 'resetting' $x$, $y$ and $z$ to 0 gives us a representation of $n$ satisfying the inequality $w>x\geq y\geq z\geq 0$ given in the theorem.

Sums of five squares

With just a few exceptions (see A047701), all positive integers can be expressed as the sum of five nonzero squares. For example, 255 = 9² + 8² + 7² + 6² + 5².

Theorem SQS5. Every integer $n>33$ can be expressed as the sum of five nonzero squares.

Proof. The cases $170>n>33$ are examined one by one in the table below, demonstrating that each of them has at least one representation as the sum of five nonzero squares. For $n>169$, we will need the 4-square representation, which may include zeroes, which Theorem SQS4 tells us exists for all $n$. What is special about 169 is not that it is a square, but that it can also be represented as the sum of five, four, three or two squares. Solve the equation $n-169=w^{2}+x^{2}+y^{2}+z^{2}$ observing the stricture $w>x\geq y\geq z\geq 0$. If $z>0$, then $n=13^{2}+w^{2}+x^{2}+y^{2}+z^{2}$. If $y>z=0$, then $n=12^{2}+5^{2}+w^{2}+x^{2}+y^{2}$. If $x>y=z=0$, then $n=12^{2}+4^{2}+3^{2}+w^{2}+x^{2}$. And if only $w>0$, then we have $n=10^{2}+8^{2}+2^{2}+1^{2}+w^{2}$. □

Of course some integers have more than one representation as a sum of five nonzero squares, and the proof of Theorem SQS5 might not always give us the most "interesting" representation. Returning to our example of 255, the method outlined in the proof would give us 255 = (12² + 5²) + (9² + 2² + 1²).

34	42 + 32 + 22 + 22 + 12	35	52 + 22 + 22 + 12 + 12	36	42 + 32 + 32 + 12 + 12
37	52 + 32 + 12 + 12 + 12	38	42 + 42 + 22 + 12 + 12	39	42 + 32 + 32 + 22 + 12
40	62 + 12 + 12 + 12 + 12	41	52 + 32 + 22 + 12 + 12	42	42 + 32 + 32 + 22 + 22
43	52 + 32 + 22 + 22 + 12	44	52 + 42 + 12 + 12 + 12	45	52 + 32 + 32 + 12 + 12
46	42 + 42 + 32 + 22 + 12	47	52 + 42 + 22 + 12 + 12	48	52 + 32 + 32 + 22 + 12
49	62 + 22 + 22 + 22 + 12	50	52 + 42 + 22 + 22 + 12	51	62 + 32 + 22 + 12 + 12
52	52 + 42 + 32 + 12 + 12	53	42 + 42 + 42 + 22 + 12	54	62 + 32 + 22 + 22 + 12
55	52 + 42 + 32 + 22 + 12	56	52 + 52 + 22 + 12 + 12	57	62 + 32 + 22 + 22 + 22
58	62 + 42 + 22 + 12 + 12	59	62 + 32 + 32 + 22 + 12	60	52 + 42 + 32 + 32 + 12
61	62 + 42 + 22 + 22 + 12	62	52 + 42 + 42 + 22 + 12	63	62 + 42 + 32 + 12 + 12
64	72 + 32 + 22 + 12 + 12	65	42 + 42 + 42 + 42 + 12	66	62 + 42 + 32 + 22 + 12
67	62 + 52 + 22 + 12 + 12	68	52 + 52 + 42 + 12 + 12	69	72 + 32 + 32 + 12 + 12
70	62 + 52 + 22 + 22 + 12	71	72 + 42 + 22 + 12 + 12	72	62 + 52 + 32 + 12 + 12
73	62 + 52 + 22 + 22 + 22	74	72 + 42 + 22 + 22 + 12	75	62 + 52 + 32 + 22 + 12
76	72 + 42 + 32 + 12 + 12	77	72 + 52 + 12 + 12 + 12	78	62 + 42 + 42 + 32 + 12
79	72 + 42 + 32 + 22 + 12	80	72 + 52 + 22 + 12 + 12	81	62 + 62 + 22 + 22 + 12
82	62 + 52 + 42 + 22 + 12	83	72 + 52 + 22 + 22 + 12	84	72 + 42 + 32 + 32 + 12
85	72 + 52 + 32 + 12 + 12	86	82 + 42 + 22 + 12 + 12	87	62 + 52 + 42 + 32 + 12
88	72 + 52 + 32 + 22 + 12	89	82 + 42 + 22 + 22 + 12	90	62 + 62 + 42 + 12 + 12
91	72 + 62 + 22 + 12 + 12	92	72 + 52 + 42 + 12 + 12	93	72 + 52 + 32 + 32 + 12
94	82 + 42 + 32 + 22 + 12	95	72 + 52 + 42 + 22 + 12	96	92 + 32 + 22 + 12 + 12
97	72 + 62 + 22 + 22 + 22	98	82 + 52 + 22 + 22 + 12	99	72 + 62 + 32 + 22 + 12
100	72 + 52 + 42 + 32 + 12	101	82 + 42 + 42 + 22 + 12	102	62 + 62 + 52 + 22 + 12
103	82 + 52 + 32 + 22 + 12	104	92 + 32 + 32 + 22 + 12	105	62 + 62 + 42 + 42 + 12
106	72 + 62 + 42 + 22 + 12	107	62 + 62 + 52 + 32 + 12	108	92 + 42 + 32 + 12 + 12
109	82 + 62 + 22 + 22 + 12	110	82 + 52 + 42 + 22 + 12	111	92 + 42 + 32 + 22 + 12
112	92 + 52 + 22 + 12 + 12	113	102 + 22 + 22 + 22 + 12	114	82 + 62 + 32 + 22 + 12
115	72 + 62 + 52 + 22 + 12	116	92 + 42 + 32 + 32 + 12	117	92 + 52 + 32 + 12 + 12
118	82 + 62 + 42 + 12 + 12	119	82 + 72 + 22 + 12 + 12	120	92 + 52 + 32 + 22 + 12
121	82 + 62 + 42 + 22 + 12	122	102 + 42 + 22 + 12 + 12	123	92 + 62 + 22 + 12 + 12
124	82 + 72 + 32 + 12 + 12	125	102 + 42 + 22 + 22 + 12	126	82 + 62 + 42 + 32 + 12
127	82 + 72 + 32 + 22 + 12	128	92 + 62 + 32 + 12 + 12	129	92 + 62 + 22 + 22 + 22
130	82 + 62 + 52 + 22 + 12	131	92 + 62 + 32 + 22 + 12	132	92 + 52 + 42 + 32 + 12
133	72 + 72 + 52 + 32 + 12	134	82 + 72 + 42 + 22 + 12	135	82 + 62 + 52 + 32 + 12
136	112 + 32 + 22 + 12 + 12	137	102 + 42 + 42 + 22 + 12	138	92 + 62 + 42 + 22 + 12
139	102 + 52 + 32 + 22 + 12	140	82 + 72 + 52 + 12 + 12	141	92 + 72 + 32 + 12 + 12
142	82 + 62 + 52 + 42 + 12	143	82 + 72 + 52 + 22 + 12	144	92 + 72 + 32 + 22 + 12
145	102 + 62 + 22 + 22 + 12	146	102 + 52 + 42 + 22 + 12	147	92 + 62 + 52 + 22 + 12
148	82 + 72 + 52 + 32 + 12	149	82 + 82 + 42 + 22 + 12	150	102 + 62 + 32 + 22 + 12
151	112 + 42 + 32 + 22 + 12	152	92 + 62 + 52 + 32 + 12	153	82 + 62 + 62 + 42 + 12
154	82 + 72 + 62 + 32 + 12	155	82 + 72 + 52 + 42 + 12	156	92 + 72 + 42 + 32 + 12
157	102 + 62 + 42 + 22 + 12	158	102 + 72 + 22 + 22 + 12	159	92 + 82 + 32 + 22 + 12
160	112 + 52 + 32 + 22 + 12	161	102 + 72 + 22 + 22 + 22	162	102 + 62 + 42 + 32 + 12
163	102 + 72 + 32 + 22 + 12	164	112 + 52 + 42 + 12 + 12	165	92 + 72 + 52 + 32 + 12
166	92 + 82 + 42 + 22 + 12	167	112 + 52 + 42 + 22 + 12	168	112 + 62 + 32 + 12 + 12
169	122 + 42 + 22 + 22 + 12

References

• Ivan Niven & Herbert S. Zuckerman, An Introduction to the Theory of Numbers, New York: John Wiley (1980).