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 arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777� ...

, 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 suf ...

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 adj ...

, 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 bicondi ...

its

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 ways ...

p \equiv 3 \pmod 4

and k is

odd Odd means unpaired, occasional, strange or unusual, or a person who is viewed as eccentric. Odd may also refer to: Acronym * ODD (Text Encoding Initiative) ("One Document Does it all"), an abstracted literate-programming format for describing X ...

.'' 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 true ar ...

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 ways ...

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 sign ...

; for instance, every Pythagorean triple

a^2+b^2=c^2

gives a second representation for

c^2

beyond the trivial representation

c^2+0^2

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 Even may refer to: General * Even (given name), a Norwegian male personal name * Even (surname) * Even (people), an ethnic group from Siberia and Russian Far East ** Even language, a language spoken by the Evens * Odd and Even, a solitaire game w ...

. 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 :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 integers; 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. The number of representable numbers in the range from 0 to any number

n

is proportional to

n/\sqrt

, 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 the ...

, approximately 0.764. The product of any two representable numbers is another representable number. Its representation can be derived from representations of its two factors, using the Brahmagupta–Fibonacci identity.

References

Additive number theory Squares in number theory Theorems in number theory {{numtheory-stub

Examples

Representable numbers

See also

References