Jacobi's four-square theorem gives a formula for the number of ways that a given positive integer n can be represented as the sum of four squares (of...

matrix Jacobi's four-square theorem, in number theory Jacobi's theorem (geometry), on concurrent lines associated with any triangle Jacobi's theorem on the...

Lagrange's four-square theorem, also known as Bachet's conjecture, states that every nonnegative integer can be represented as a sum of four non-negative...

short proof derived from the Jacobi triple product. Legendre's three-square theorem Lagrange's four-square theorem Sum of squares function Brahmagupta–Fibonacci...

four square Four-square cipher Lagrange's four-square theorem, stating that any natural number equals the sum of four integers squared Jacobi's four-square...

four-square theorem Legendre's three-square theorem states which numbers can be expressed as the sum of three squares Jacobi's four-square theorem gives...

Jacob Jacobi and published in its Latinized form as Carolus Gustavus Jacobus Jacobi. He is sometimes referred to as C. G. J. Jacobi. One of Jacobi's greatest...

Legendre's three-square theorem and Jacobi's four-square theorem; and in statistics, the analysis of variance involves summing the squares of quantities...

irreducibility theorem (number theory) Hurwitz's theorem (number theory) Jacobi's four-square theorem (number theory) Jurkat–Richert theorem (analytic number...

partition Jacobi's four-square theorem Gauss circle problem P. T. Bateman (1951). "On the Representation of a Number as the Sum of Three Squares" (PDF)....

field Jacobi's four-square theorem Jacobi form Jacobi's formula Jacobi group Jacobian ideal Jacobi identity Jacobi integral Jacobi's logarithm Jacobi method...

(1859) on the zeta function is the canonical starting point; Jacobi's four-square theorem (1839), which predates it, belongs to an initially different...

1{\bmod {4}}} be a prime not equal to p {\displaystyle p} . By Jacobi's four-square theorem, there are p + 1 {\displaystyle p+1} solutions to the equation...

the full set. (This notation is due to Gudermann and Glaisher and is not Jacobi's original notation.) Throughout this article, pq ⁡ ( u , t 2 ) = pq ⁡ (...

In mechanics, the virial theorem provides a general equation that relates the average over time of the total kinetic energy of a stable system of discrete...

of Fermat's theorem on sums of two squares, which states that an odd prime ⁠ p {\displaystyle p} ⁠ is expressible as the sum of two squares, ⁠ p = x 2...

square number, while other divisors come in pairs. Lagrange's four-square theorem states that any positive integer can be written as the sum of four or...

{\displaystyle \mathbf {M} } ⁠ being a normal matrix, and thus also square, the spectral theorem ensures that it can be unitarily diagonalized using a basis of...

In number theory, the law of quadratic reciprocity is a theorem about modular arithmetic that gives conditions for the solvability of quadratic equations...

_{k=1}^{n}k\right)^{2}.} This identity is sometimes called Nicomachus's theorem, after Nicomachus of Gerasa (c. 60 – c. 120 CE). Nicomachus, at the end...

Factorization RSA number Fundamental theorem of arithmetic Square-free Square-free integer Square-free polynomial Square number Power of two Integer-valued...

theorem Euler spiral – a curve whose curvature varies linearly with its arc length Euler squares, usually called Graeco-Latin squares Euler's theorem...

mathematician. In number theory, he proved special cases of Fermat's Last Theorem and created analytic number theory. In analysis, he advanced the theory...

\chi \operatorname {sn} \psi ,\end{aligned}}} where sn, cn and dn are Jacobi's elliptic functions. The Laplace–Runge–Lenz vector can also be generalized...

forms of Fermat's right triangle theorem.: p. 14  Each primitive Pythagorean triangle has a ratio of area, K, to squared semiperimeter, s, that is unique...

follows Riemann and Mumford; Jacobi's original formulation was in terms of the nome q = eiπτ rather than τ. In Jacobi's notation the θ-functions are written:...

of Jacobi's two-square theorem, which follows almost immediately from the Jacobi triple product. A much simpler sum appears if the sum of squares function...

polynomial are, by the Abel–Ruffini theorem, the highest degree equations having a general solution using radicals. Square (algebra) Cube (algebra) Exponentiation...

S, and the diagonal entries of B are uniquely determined – this is Jacobi's theorem. If S is allowed to be any invertible matrix then B can be made to...

The other solution involves the logarithm function. Bochner's theorem states that the Jacobi polynomials are uniquely characterized as polynomial solutions...