Jacobi's four-square theorem gives a formula for the number of ways that a given positive integer ''n'' can be represented as the sum of four squares.
History
The theorem was proved in 1834 by
Carl Gustav Jakob Jacobi
Carl Gustav Jacob Jacobi (; ; 10 December 1804 – 18 February 1851) was a German mathematician who made fundamental contributions to elliptic functions, dynamics, differential equations, determinants, and number theory. His name is occasiona ...
.
Theorem
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:
:
The number of ways to represent n as the sum of four squares is eight times the sum of the
divisor
In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a multiple of m. An integer n is divisible or evenly divisible by ...
s of ''n'' if ''n'' is odd and 24 times the sum of the odd divisors of ''n'' if ''n'' is even (see
divisor function
In mathematics, and specifically in number theory, a divisor function is an arithmetic function related to the divisors of an integer. When referred to as ''the'' divisor function, it counts the ''number of divisors of an integer'' (including ...
), i.e.
:
Equivalently, it is eight times the sum of all its divisors which are not divisible by 4, i.e.
:
We may also write this as
:
where the second term is to be taken as zero if ''n'' is not divisible by 4. In particular, for a
prime number
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
''p'' we have the explicit formula ''r''
4(''p'') = 8(''p'' + 1).
[.]
Some values of ''r''
4(''n'') occur infinitely often as ''r''
4(''n'') = ''r''
4(2
''m''''n'') whenever ''n'' is even. The values of ''r''
4(''n'') can be arbitrarily large: indeed, ''r''
4(''n'') is infinitely often larger than 8.
Proof
The theorem can be proved by elementary means starting with the
Jacobi triple product
In mathematics, the Jacobi triple product is the mathematical identity:
:\prod_^\infty
\left( 1 - x^\right)
\left( 1 + x^ y^2\right)
\left( 1 +\frac\right)
= \sum_^\infty x^ y^,
for complex numbers ''x'' and ''y'', with , ''x'', < 1 and ''y ...
.
The proof shows that the
Theta series
In mathematics, theta functions are special functions of several complex variables. They show up in many topics, including Abelian varieties, moduli spaces, quadratic forms, and solitons. As Grassmann algebras, they appear in quantum field theo ...
for the
lattice
Lattice may refer to:
Arts and design
* Latticework, an ornamental criss-crossed framework, an arrangement of crossing laths or other thin strips of material
* Lattice (music), an organized grid model of pitch ratios
* Lattice (pastry), an orna ...
Z
4 is a
modular form
In mathematics, a modular form is a (complex) analytic function on the upper half-plane satisfying a certain kind of functional equation with respect to the Group action (mathematics), group action of the modular group, and also satisfying a grow ...
of a certain level, and hence equals a
linear combination of
Eisenstein series
Eisenstein series, named after German mathematician Gotthold Eisenstein, are particular modular forms with infinite series expansions that may be written down directly. Originally defined for the modular group, Eisenstein series can be generaliz ...
.
See also
*
Lagrange's four-square theorem
Lagrange's four-square theorem, also known as Bachet's conjecture, states that every natural number can be represented as the sum of four integer squares. That is, the squares form an additive basis of order four.
p = a_0^2 + a_1^2 + a_2^2 + a_ ...
*
Lambert series
In mathematics, a Lambert series, named for Johann Heinrich Lambert, is a series taking the form
:S(q)=\sum_^\infty a_n \frac .
It can be resumed formally by expanding the denominator:
:S(q)=\sum_^\infty a_n \sum_^\infty q^ = \sum_^\infty b_m ...
*
Sum of squares function 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 sign ...
Notes
References
*
*
*
External links
*{{MathWorld, id=SumofSquaresFunction, title=Sum of Squares Function
Squares in number theory
Theorems in number theory