Jacobi's four-square theorem
- Jacobi's four-square 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.
Equivalently, it is eight times the sum of all its divisors which are not divisible by 4, i.e.
In particular, for a prime number "p" we have the explicit formula .
See also
*Lagrange's four-square theorem
*Lambert series
References
*cite journal|first=Michael D.|last=Hirschhorn|coauthors=James A. Mcgowan|title=Algebraic consequences of Jacobi’s two– and four–square theorems|url=http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.26.9028|journal=Ismail (eds), Developments in Mathematics|pages=107-132
*cite journal|first=Michael D.|last=Hirschhorn|title=A simple proof of Jacobi’s four-square theorem|date=1987|journal=Proc. Amer. Math. Soc
External links
* [http://www.math.ohio-state.edu/~econrad/Jacobi/sumofsq/sumofsq.html Eric Conrad's page]
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