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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 .
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.
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.
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.
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 ...
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.
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.
Jacobi's theorem can refer to: Maximum power theorem, in electrical engineering; The result that the determinant of skew-symmetric matrices with odd size vanishes, see skew-symmetric matrix; Jacobi's four-square theorem, in number theory; Jacobi's theorem (geometry), on concurrent lines associated with any triangle