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Lagrange's four-square theorem can be refined in various ways. For example, Zhi-Wei Sun [ 14 ] proved that each natural number can be written as a sum of four squares with some requirements on the choice of these four numbers.
In mathematics, Lagrange's theorem usually refers to any of the following theorems, attributed to Joseph Louis Lagrange: Lagrange's theorem (group theory) Lagrange's theorem (number theory) Lagrange's four-square theorem, which states that every positive integer can be expressed as the sum of four squares of integers; Mean value theorem in calculus
Euler proved Newton's identities, Fermat's little theorem, Fermat's theorem on sums of two squares, and made distinct contributions to the Lagrange's four-square theorem. He also invented the totient function φ(n) which assigns to a positive integer n the number of positive integers less than n and coprime to n.
Joseph-Louis Lagrange [a] (born Giuseppe Luigi Lagrangia [5] [b] or Giuseppe Ludovico De la Grange Tournier; [6] [c] 25 January 1736 – 10 April 1813), also reported as Giuseppe Luigi Lagrange [7] or Lagrangia, [8] was an Italian mathematician, physicist and astronomer, later naturalized French.
Lagrange's four-square theorem says that the set of positive square numbers is an additive basis of order 4. Another highly non-trivial and celebrated result along these lines is Vinogradov's theorem. One is naturally inclined to ask whether these results are optimal.
Lagrange's four-square theorem of 1770 states that every natural number is the sum of at most four squares. Since three squares are not enough, this theorem establishes g ( 2 ) = 4 {\displaystyle g(2)=4} .
Jacobi's four-square theorem (number theory) Jurkat–Richert theorem (analytic number theory) Kaplansky's theorem on quadratic forms (number theory) Khinchin's theorem (probability) Kronecker's theorem (Diophantine approximation) Kronecker–Weber theorem (number theory) Lafforgue's theorem (algebraic number theory) Lagrange's four-square ...
Legendre's three-square theorem states which numbers can be expressed as the sum of three squares; Jacobi's four-square theorem gives the number of ways that a number can be represented as the sum of four squares. For the number of representations of a positive integer as a sum of squares of k integers, see Sum of squares function. Fermat's ...