Jacobi's four-square theorem

The theorem was proved in 1834 by Carl Gustav Jakob Jacobi.

Two representations are considered different if their terms are in different order or if the integer being squared (not just the square) is different; to illustrate, these are three of the eight different ways to represent 1:

$$\backslash begin\{align\}$$

1^2 &+ 0^2 + 0^2 + 0^2 \\

0^2 &+ 1^2 + 0^2 + 0^2 \\

(-1)^2 &+ 0^2 + 0^2 + 0^2.

\end{align}

The number of ways to represent {{mvar|n}} as the sum of four squares is eight times the sum of the divisors of {{mvar|n}} if {{mvar|n}} is odd and 24 times the sum of the odd divisors of {{mvar|n}} if {{mvar|n}} is even (see divisor function), i.e.

$$r\_4(n)\; =\; \backslash begin\{cases\}$$

\displaystyle 8\sum_{m|n} m & \text{if } n \text{ is odd}, \\[12pt]

\displaystyle 24 \sum_{{m|n} \atop {m\text{ odd}}} m & \text{if } n \text{ is even}.

\end{cases}

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

$$r\_4(n)\; =\; 8\; \backslash sum\_\{\{m\; \backslash mid\; n,\}\; \backslash atop\; \{4\; \backslash nmid\; m\}\}\; m.$$

An immediate consequence is $r\_4(2n)\; =\; r\_4(8n)$; for odd $n$, $r\_4(2\; \backslash cdot\; 4^a\; n)\; =\; r\_4(2\; n)$.{{Cite book |last=Grosswald |first=Emil |url=https://archive.org/details/representationso0000gros |title=Representations of integers as sums of squares |date=1985 |publisher=New York : Springer-Verlag |others=Internet Archive |isbn=978-0-387-96126-2 |pages=30}}

We may also write this as

$$r\_4(n)\; =\; 8\; \backslash ,\; \backslash sigma(n)\; -32\; \backslash ,\; \backslash sigma(n/4)$$

where the second term is to be taken as zero if {{mvar|n}} is not divisible by 4. In particular, for a prime number {{mvar|p}} we have the explicit formula {{math|1=r₄(p) = 8(p + 1)}}.{{harvnb|Williams|2011|p=119}}.

Some values of {{math|r₄(n)}} occur infinitely often as {{math|1=r₄(n) = r₄(2^mn)}} whenever {{mvar|n}} is even. The values of {{math|r₄(n)}} can be arbitrarily large: indeed, {{math|r₄(n)}} is infinitely often larger than $8\backslash sqrt\{\backslash log\; n\}.$

The theorem can be proved by elementary means starting with the Jacobi triple product.{{Cite journal |last=Hirschhorn |first=Michael D. |date=2000 |title=Partial Fractions and Four Classical Theorems of Number Theory |journal=The American Mathematical Monthly |volume=107 |issue=3 |pages=260–264 |citeseerx=10.1.1.28.1615 |doi=10.2307/2589321 |jstor=2589321 }}

The proof shows that the Theta series for the lattice Z⁴ is a modular form of a certain level, and hence equals a linear combination of Eisenstein series.

• Lagrange's four-square theorem
• Lambert series
• Sum of squares function

{{reflist}}

• {{cite book |last1 = Hirschhorn |first1 = Michael D. |last2 = McGowan |first2 = James A. |year = 2001 |chapter = Algebraic Consequences of Jacobi's Two— and Four—Square Theorems |editor-last1=Garvan |editor-first1=F. G. |editor-last2=Ismail |editor-first2=M. E. H. |title = Symbolic Computation, Number Theory, Special Functions, Physics and Combinatorics |series = Developments in Mathematics |volume = 4 |pages = 107–132 |publisher=Springer |citeseerx=10.1.1.26.9028 |doi = 10.1007/978-1-4613-0257-5_7 |isbn = 978-1-4020-0101-7}}
• {{cite journal |last=Hirschhorn |first=Michael D. |title=A simple proof of Jacobi's four-square theorem |date=1987 |journal= Proceedings of the American Mathematical Society|volume=101 |issue=3 |pages=436–438 |doi=10.1090/s0002-9939-1987-0908644-9|doi-access=free }}
• {{cite book | last=Williams | first=Kenneth S. | year=2011 | title=Number theory in the spirit of Liouville | series=London Mathematical Society Student Texts | volume=76 | publisher=Cambridge University Press | isbn=978-0-521-17562-3 | zbl=1227.11002 }}

• {{MathWorld|id=SumofSquaresFunction|title=Sum of Squares Function}}

Category:Squares in number theory

Category:Theorems in number theory