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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.
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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.