In number theory, the sum of squares function is an arithmetic function that gives the number of representations for a given positive integer as the sum of squares, where representations that differ only in the order of the summands or in the signs of the numbers being squared are counted as different. It is denoted by .
Definition The function is defined as
where denotes the cardinality of a set. In other words, is the number of ways can be written as a sum of squares.
For example, since where each sum has two sign combinations, and also since with four sign combinations. On the other hand, because there is no way to represent 3 as a sum of two squares.
Formulae
k = 2
The number of ways to write a natural number as sum of two squares is given by . It is given explicitly by
where is the number of divisors of which are congruent to 1 modulo 4 and is the number of divisors of which are congruent to 3 modulo 4. Using sums, the expression can be written as:
The prime factorization , where are the prime factors of the form and are the prime factors of the form gives another formula , if all exponents are even. If one or more are odd, then .
k = 3
Gauss proved that for a squarefree number ,
where denotes the class number of an integer .
There exist extensions of Gauss' formula to arbitrary integer .
k = 4
The number of ways to represent as the sum of four squares was due to Carl Gustav Jakob Jacobi and it is eight times the sum of all its divisors which are not divisible by 4, i.e.
Representing , where is an odd integer, one can express in terms of the divisor function as follows:
k = 6
The number of ways to represent as the sum of six squares is given by
where is the Kronecker symbol.
k = 8 Jacobi also found an explicit formula for the case :
Generating function The generating function of the sequence for fixed can be expressed in terms of the Jacobi theta function:
where
Numerical values The first 30 values for are listed in the table below: n = 0 0 1 1 1 1 1 1 1 1 1 1 2 4 6 8 10 12 14 16 2 2 0 4 12 24 40 60 84 112 3 3 0 0 8 32 80 160 280 448 4 22 2 4 6 24 90 252 574 1136 5 5 0 8 24 48 112 312 840 2016 6 2×3 0 0 24 96 240 544 1288 3136 7 7 0 0 0 64 320 960 2368 5504 8 23 0 4 12 24 200 1020 3444 9328 9 32 2 4 30 104 250 876 3542 12112 10 2×5 0 8 24 144 560 1560 4424 14112 11 11 0 0 24 96 560 2400 7560 21312 12 22×3 0 0 8 96 400 2080 9240 31808 13 13 0 8 24 112 560 2040 8456 35168 14 2×7 0 0 48 192 800 3264 11088 38528 15 3×5 0 0 0 192 960 4160 16576 56448 16 24 2 4 6 24 730 4092 18494 74864 17 17 0 8 48 144 480 3480 17808 78624 18 2×32 0 4 36 312 1240 4380 19740 84784 19 19 0 0 24 160 1520 7200 27720 109760 20 22×5 0 8 24 144 752 6552 34440 143136 21 3×7 0 0 48 256 1120 4608 29456 154112 22 2×11 0 0 24 288 1840 8160 31304 149184 23 23 0 0 0 192 1600 10560 49728 194688 24 23×3 0 0 24 96 1200 8224 52808 261184 25 52 2 12 30 248 1210 7812 43414 252016 26 2×13 0 8 72 336 2000 10200 52248 246176 27 33 0 0 32 320 2240 13120 68320 327040 28 22×7 0 0 0 192 1600 12480 74048 390784 29 29 0 8 72 240 1680 10104 68376 390240 30 2×3×5 0 0 48 576 2720 14144 71120 395136
See also
Integer partition Jacobi's four-square theorem Gauss circle problem
References
Further reading
External links