In
number theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Math ...
, the sum of two squares theorem relates the
prime decomposition
In number theory, integer factorization is the decomposition of a composite number into a product of smaller integers. If these factors are further restricted to prime numbers, the process is called prime factorization.
When the numbers are su ...
of any
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
to whether it can be written as a sum of two
squares
In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90-degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length ad ...
, such that for some integers , .
:''An integer greater than one can be written as a sum of two squares
if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is bi ...
its
prime decomposition
In number theory, integer factorization is the decomposition of a composite number into a product of smaller integers. If these factors are further restricted to prime numbers, the process is called prime factorization.
When the numbers are su ...
contains no factor , where
prime
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 way ...
and k is
odd.''
In writing a number as a sum of two squares, it is allowed for one of the squares to be zero, or for both of them to be equal to each other, so all squares and all doubles of squares are included in the numbers that can be represented in this way. This theorem supplements
Fermat's theorem on sums of two squares
In additive number theory, Fermat's theorem on sums of two squares states that an odd prime ''p'' can be expressed as:
:p = x^2 + y^2,
with ''x'' and ''y'' integers, if and only if
:p \equiv 1 \pmod.
The prime numbers for which this is tr ...
which says when 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 way ...
can be written as a sum of two squares, in that it also covers the case for
composite number
A composite number is a positive integer that can be formed by multiplying two smaller positive integers. Equivalently, it is a positive integer that has at least one divisor other than 1 and itself. Every positive integer is composite, prime, ...
s.
A number may have multiple representations as a sum of two squares, counted by the
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 s ...
; for instance, every
Pythagorean triple
A Pythagorean triple consists of three positive integers , , and , such that . Such a triple is commonly written , and a well-known example is . If is a Pythagorean triple, then so is for any positive integer . A primitive Pythagorean triple is ...
gives a second representation for
beyond the trivial representation
.
Examples
The prime decomposition of the number 2450 is given by 2450 = 257. Of the primes occurring in this decomposition, 2, 5, and 7, only 7 is congruent to 3 modulo 4. Its exponent in the decomposition, 2, is
even. Therefore, the theorem states that it is expressible as the sum of two squares. Indeed, .
The prime decomposition of the number 3430 is 257. This time, the exponent of 7 in the decomposition is 3, an odd number. So 3430 cannot be written as the sum of two squares.
Representable numbers
The numbers that can be represented as the sums of two squares form the
integer sequences
In mathematics, an integer sequence is a sequence (i.e., an ordered list) of integers.
An integer sequence may be specified ''explicitly'' by giving a formula for its ''n''th term, or ''implicitly'' by giving a relationship between its terms. Fo ...
:0, 1, 2, 4, 5, 8, 9, 10, 13, 16, 17, 18, 20, 25, 26, 29, 32, ...
They form the set of all
norms of
Gaussian integer
In number theory, a Gaussian integer is a complex number whose real and imaginary parts are both integers. The Gaussian integers, with ordinary addition and multiplication of complex numbers, form an integral domain, usually written as \mathbf /ma ...
s;
[ their square roots form the set of all lengths of ]line segment
In geometry, a line segment is a part of a straight line that is bounded by two distinct end points, and contains every point on the line that is between its endpoints. The length of a line segment is given by the Euclidean distance between ...
s between pairs of points in the two-dimensional integer lattice
In mathematics, the -dimensional integer lattice (or cubic lattice), denoted , is the lattice in the Euclidean space whose lattice points are -tuples of integers. The two-dimensional integer lattice is also called the square lattice, or gri ...
.
The number of representable numbers in the range from 0 to any number is proportional to , with a limiting constant of proportionality given by the Landau–Ramanujan constant In mathematics and the field of number theory, the Landau–Ramanujan constant is the positive real number ''b'' that occurs in a theorem proved by Edmund Landau in 1908, stating that for large x, the number of positive integers below x that are th ...
, approximately 0.764.Brahmagupta–Fibonacci identity
In algebra, the Brahmagupta–Fibonacci identity expresses the product of two sums of two squares as a sum of two squares in two different ways. Hence the set of all sums of two squares is closed under multiplication. Specifically, the identity sa ...
.
See also
* Legendre's three-square theorem
* 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_3 ...
* 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 s ...
References
Additive number theory
Squares in number theory
Theorems in number theory
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