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The theorem can be proved by elementary means starting with the Jacobi triple product. [ 3 ] The proof shows that the Theta series for the lattice Z 4 is a modular form of a certain level, and hence equals a linear combination of Eisenstein series .
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 theorem on sums of two squares says which primes are sums of two squares.
The use of complex analysis in number theory comes later: the work of Bernhard Riemann (1859) on the zeta function is the canonical starting point; [77] Jacobi's four-square theorem (1839), which predates it, belongs to an initially different strand that has by now taken a leading role in analytic number theory (modular forms). [78]
Jacobi's four-square theorem; K. Kaplansky's theorem on quadratic forms; Katz–Lang finiteness theorem; Kronecker's congruence; Kummer's congruence; Kummer's theorem; L.
Jacobi's four-square theorem; L. Lagrange's four-square theorem; Legendre's conjecture; Legendre's three-square theorem; P. Pythagorean prime; Pythagorean quadruple;
Later, in 1834, Carl Gustav Jakob Jacobi discovered a simple formula for the number of representations of an integer as the sum of four squares with his own four-square theorem. The formula is also linked to Descartes' theorem of four "kissing circles", which involves the sum of the squares of the curvatures of four circles.
Carl Gustav Jacob Jacobi. Jacobi was the first to apply elliptic functions to number theory, for example proving Fermat's two-square theorem and Lagrange's four-square theorem, and similar results for 6 and 8 squares.
In mathematics and statistics, sums of powers occur in a number of contexts: . Sums of squares arise in many contexts. For example, in geometry, the Pythagorean theorem involves the sum of two squares; in number theory, there are Legendre's three-square theorem and Jacobi's four-square theorem; and in statistics, the analysis of variance involves summing the squares of quantities.