Lagrange's four-square theorem

Statement

The statement has the following equivalent forms:

1. Every nonnegative integer can be expressed as a sum of four squares of integers.
2. Every nonnegative integer can be expressed as a sum of four squares of nonnegative integers. Note that one or more of the integers is allowed to be zero.
3. Every positive integer is a sum of at most four perfect squares.

Interpretations

Finite algebraic characterization of nonnegative integers

Lagrange's four-square theorem gives a characterization of the nonnegative integers using a finite formula purely in terms of the ring theoretic properties of the integers.

Related facts

Identities used/involved

• Brahmagupta-Fibonacci two-square identity
• Euler's four-square identity

Similar facts about sums of squares

• Jacobi's four-square theorem gives an explicit formula for the number of ways a given nonnegative integer can be expressed as a sum of four squares.
• Fermat's theorem on sums of two squares characterizes those nonnegative integers that can be expressed as sums of two squares.
• Gauss's three-square theorem characterizes those nonnegative integers that can be expressed as a sum of three squares.

Questions and facts about sums of higher powers

• Waring's problem asks to find, for a given $d$, the smallest number $g(d)$ such that every nonnegative integer can be expressed as a sum of $g(d)$ $d^{th}$ powers of nonnegative integers.
• Generalized Waring problem asks to find, for a given $d$, the smallest number $G(d)$ such that every nonnegative integer can be expressed as a sum of $g(d)$ $d^{th}$ powers of nonnegative integers.