Lagrange's four-square theorem
Every natural number can be represented as the sum of four integer squares / From Wikipedia, the free encyclopedia
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For Lagrange's identity, see Lagrange's identity (disambiguation). For Lagrange's theorem, see Lagrange's theorem (disambiguation).
"four-square theorem" and "four square theorem" redirect here. For other uses, see four square (disambiguation).
Lagrange's four-square theorem, also known as Bachet's conjecture, states that every natural number can be represented as a sum of four non-negative integer squares.[1] That is, the squares form an additive basis of order four.
where the four numbers are integers. For illustration, 3, 31, and 310 in several ways, can be represented as the sum of four squares as follows:
This theorem was proven by Joseph Louis Lagrange in 1770. It is a special case of the Fermat polygonal number theorem.