mathematics, Euler's four-square identity says that the product of two numbers, each of which is a sum of four squares, is itself a sum of four squares. For any...

for every odd prime number p. This immediately follows from Euler's four-square identity (and from the fact that the theorem is true for the numbers 1...

integral Euler–Riemann zeta function Euler's identity e iπ + 1 = 0. Euler's four-square identity, which shows that the product of two sums of four squares can...

The identity is also known as the Diophantus identity, as it was first proved by Diophantus of Alexandria. It is a special case of Euler's four-square identity...

Catalan identities Degen's eight-square identity Difference of two squares Euler's four-square identity Euler's identity Fibonacci's identity see Brahmagupta–Fibonacci...

power Euler's four-square identity – Product of sums of four squares expressed as a sum of four squares Fermat's theorem on sums of two squares – Condition...

Difference of two squares Euler's four-square identity, related to quaternions in the same way Lagrange's identity Other Parseval's identity Pythagorean trigonometric...

to this work included Euler's four-square identity (1748) and Olinde Rodrigues' parameterization of general rotations by four parameters (1840), but...

Jacobi's four-square theorem giving the number of distinct ways an integer can be represented as the sum of four squares Euler's four-square identity or theorem...

tedious, to check that a2 + b2 + c2 + d2 = 1. (This is essentially Euler's four-square identity.) Any central rotation in three dimensions is uniquely determined...

{\displaystyle C=-2} in hyperbolic geometry. Circle packing in a circle Euler's four-square identity Malfatti circles Soddy, F. (June 1936), "The Kiss Precise", Nature...

Poincaré group Generalizations of Pauli matrices Bloch sphere Euler's four-square identity For higher spin generalizations of the Pauli matrices, see Spin...

statements are true for quaternions (Euler's four-square identity), complex numbers (the Brahmagupta–Fibonacci two-square identity) and real numbers. In 1898 Adolf...

of Euler's four-square and Degen's eight-square identities. Brahmagupta–Fibonacci identity Euler's four-square identity Degen's eight-square identity Sedenions...

theorem (very compact proof) Erdős–Ko–Rado theorem Euler's formula Euler's four-square identity Euler's theorem Five color theorem Five lemma Fundamental...

a sum of two squares is divisible by a prime which is a sum of two squares, then the quotient is a sum of two squares. (This is Euler's first Proposition)...

Eruditorum, 1744 The title page of Euler's Methodus inveniendi lineas curvas Euler's 1760 world map Euler's 1753 map of Africa Euler is listed by an academic genealogy...

logarithm, also commonly written as ln(x) or loge(x). Euler's constant (sometimes called the Euler–Mascheroni constant) is a mathematical constant, usually...

Proofs of Fermat's little theorem Fermat quotient Euler's totient function Noncototient Nontotient Euler's theorem Wilson's theorem Primitive root modulo...

prime numbers are distributed. Euler's work in this area led to the development of the prime number theorem. Euler's great interest in number theory...

(known as the Brahmagupta–Fibonacci identity), and also Euler's four-square identity and Degen's eight-square identity. These may be interpreted as multiplicativity...

before the end of 1843, Graves successfully extended to eight squares Euler's four-square identity, and went on to conceive a theory of "octaves" (now called...

identity in calculus Lagrange's trigonometric identities, two trigonometric identities Lagrange's four-square theorem, a theorem from number theory Lagrange...

sets of four integers such that the sum of the squares of the first three equals the square of the fourth. The Basel problem, solved by Euler in terms...

third the length of the longest median of the triangle. By Euler's theorem in geometry, the squared distance from the incenter I to the circumcenter O is given...

unit complex numbers under multiplication. Written as a special case of Euler's formula for an integer n, i n = exp ( 1 2 π i ) n = exp ( 1 2 n π i ) =...

way that is similar to that of the above proof of Euler's identity. One can also use Euler's identity for expressing all trigonometric functions in terms...

{3}})^{2}}}(a^{2}+b^{2}+c^{2}+d^{2})} with equality only for a square. A corollary to Euler's quadrilateral theorem is the inequality a 2 + b 2 + c 2 + d...

Euler's number e {\displaystyle e} , shown by Charles Hermite in 1873, with Euler's identity e i π = − 1. {\displaystyle e^{i\pi }=-1.} This identity...

Stegun 1972, p. 73, 4.3.45 Wu, Rex H. "Proof Without Words: Euler's Arctangent Identity", Mathematics Magazine 77(3), June 2004, p. 189. S. M. Abrarov;...