In
number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example ...
, Jacobi's four-square theorem gives a formula for the number of ways that a given
positive integer
In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positiv ...
can be represented as
the sum of four
squares
In geometry, a square is a regular polygon, regular quadrilateral. It has four straight sides of equal length and four equal angles. Squares are special cases of rectangles, which have four equal angles, and of rhombuses, which have four equal si ...
(of
integer
An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...
s).
History
The theorem was proved in 1834 by
Carl Gustav Jakob Jacobi.
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 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 divisibl ...
s of if is odd and 24 times the sum of the odd divisors of if 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'' (includi ...
), i.e.
Equivalently, it is eight times the sum of all its divisors which are not divisible by 4, i.e.
An immediate consequence is
; for odd
,
.
We may also write this as
where the second term is to be taken as zero if 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 (mathematics), 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 ...
we have the explicit formula .
[.]
Some values of occur infinitely often as whenever is even. The values of can be arbitrarily large: indeed, is infinitely often larger than
Proof
The theorem can be proved by elementary means starting with the
Jacobi triple product.
The proof shows that the
Theta series for the
lattice Z
4 is a
modular form
In mathematics, a modular form is a holomorphic function on the complex upper half-plane, \mathcal, that roughly satisfies a functional equation with respect to the group action of the modular group and a growth condition. The theory of modul ...
of a certain level, and hence equals a
linear combination
In mathematics, a linear combination or superposition is an Expression (mathematics), expression constructed from a Set (mathematics), set of terms by multiplying each term by a constant and adding the results (e.g. a linear combination of ''x'' a ...
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 generalize ...
.
See also
*
Lagrange's four-square theorem
Lagrange's four-square theorem, also known as Bachet's conjecture, states that every natural number, nonnegative integer can be represented as a sum of four non-negative integer square number, squares. That is, the squares form an additive basi ...
*
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 resummed formally by expanding the denominator:
:S(q)=\sum_^\infty a_n \sum_^\infty q^ = \sum_^\infty ...
*
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 si ...
Notes
References
*
*
*
External links
*{{MathWorld, id=SumofSquaresFunction, title=Sum of Squares Function
Squares in number theory
Theorems in number theory