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2010-08-13T22:16:19Z

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==Statement==

The statement has the following equivalent forms:

# Every nonnegative integer can be expressed as a [[fact about::sum of four squares]] of integers.
# 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.
# Every positive integer is a sum of ''at most'' four [[fact about::perfect square]]s.

==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 <math>d</math>, the smallest number <math>g(d)</math> such that every nonnegative integer can be expressed as a sum of <math>g(d)</math> <math>d^{th}</math> powers of nonnegative integers.
* [[Generalized Waring problem]] asks to find, for a given <math>d</math>, the smallest number <math>G(d)</math> such that every nonnegative integer can be expressed as a sum of <math>g(d)</math> <math>d^{th}</math> powers of nonnegative integers.

Lagrange's four-square theorem - Revision history

Vipul: Created page with "==Statement== The statement has the following equivalent forms: # Every nonnegative integer can be expressed as a fact about::sum of four squares of integers. # Every nonne..."