Jacobi's four-square theorem

For other theorems of Jacobi see Jacobi's theorem.

In 1834, Carl Gustav Jakob Jacobi found an exact formula for the total number of ways a given positive integer n can be represented as the sum of four squares. This number is eight times the sum of the divisors of n if n is odd and 24 times the sum of the odd divisors of n if n is even (see divisor function), i.e.

$r_{4}(n)={\begin{cases}8\sum _{m|n}m&{\mbox{if }}n{\mbox{ is odd}}\\24\sum _{m|n,m{\text{ odd}}}m&{\mbox{if }}n{\mbox{ is even}}.\end{cases}}$

Equivalently, it is eight times the sum of all its divisors which are not divisible by 4, i.e.

$r_{4}(n)=8\sum _{m|n,4\nmid m}m.$

In particular, for a prime number p we have the explicit formula $r_{4}(p)=8(p+1)$.

See also

• Lagrange's four-square theorem
• Lambert series

References

• Hirschhorn, Michael D. "Algebraic consequences of Jacobi's two– and four–square theorems". Ismail (eds), Developments in Mathematics: 107–132. {{cite journal}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)
• Hirschhorn, Michael D. (1987). "A simple proof of Jacobi's four-square theorem". Proc. Amer. Math. Soc.

External links

• Eric Conrad's page