In number theory, Euler's criterion is a formula for determining whether an integer is a quadratic residue modulo a prime. Precisely, Let p be an odd...

been given simple yet ambiguous names such as Euler's function, Euler's equation, and Euler's formula. Euler's work touched upon so many fields that he is...

are a rearrangement of the latter. This method can also be used to prove Euler's theorem, with a slight alteration in that the numbers from 1 to p − 1 are...

are (p + 1)/2 residues (including 0) and (p − 1)/2 nonresidues, by Euler's criterion. In this case, it is customary to consider 0 as a special case and...

in the case p ≡ 3 mod 4 {\displaystyle p\equiv 3{\bmod {4}}} using Euler's criterion one can give an explicit formula for the "square roots" modulo p {\displaystyle...

xi gives Euler's theorem: a φ ( n ) ≡ 1 ( mod n ) . {\displaystyle a^{\varphi (n)}\equiv 1{\pmod {n}}.} Carmichael number Euler's criterion Wilson's theorem...

little Fermat theorem", Acta Arithmetica, 12: 355–364, MR 0213289 Euler's criterion Pocklington test on Mathworld P. Erdős; C. Pomerance (1986). "On the...

divide a, then ap−1 ≡ 1 (mod p). Euler's theorem: If a and m are coprime, then aφ(m) ≡ 1 (mod m), where φ is Euler's totient function. A simple consequence...

and a prime p > 2 {\displaystyle p>2} (which will always be odd), Euler's criterion tells us that n {\displaystyle n} has a square root (i.e., n {\displaystyle...

an odd k where k < 2n – and if there exists an integer a for which Euler's criterion yields –1, that is, a p − 1 2 ≡ − 1 ( mod p ) , {\displaystyle a^{\frac...

{\text{ and }}\quad \left({\frac {a}{p}}\right)\in \{-1,0,1\}.} By Euler's criterion, which had been discovered earlier and was known to Legendre, these...

In continuum mechanics, the maximum distortion energy criterion (also von Mises yield criterion) states that yielding of a ductile material begins when...

{\displaystyle m^{2}+1} is divisible by p (we can also see this by Euler's criterion); it also uses the fact that the Gaussian integers are a unique factorization...

different ways, one using Euler's Criterion and the other using the Binomial theorem. As an example of how Euler's criterion is used, we can use it to...

{p-1}{2}}{\pmod {p}}.} Euler's theorem Euler's theorem states that if n and a are coprime positive integers, then aφ(n) is congruent to 1 mod n. Euler's theorem generalizes...

actually written by Johann Christian Lange, rather than Weise. He references Euler's Letters to a German Princess. In Hamilton's illustration of the four categorical...

congruence 2φ(n) ≡ 1 (mod n2), where φ denotes the Euler's totient function (according to Euler's theorem, 2φ(n) ≡ 1 (mod n) for every odd natural number...

is a quadratic nonresidue modulo M p . {\displaystyle M_{p}.} By Euler's criterion, this is equivalent to 3 M p − 1 2 ≡ − 1 ( mod M p ) . {\displaystyle...

{1}{2}}(p-1)}\equiv c\cdot 1\mod p} The last step is justified by Euler's criterion. As an example, take p = 7 {\displaystyle p=7} and q = 11 {\displaystyle...

There is another way the Jacobi and Legendre symbols differ. If the Euler's criterion formula is used modulo a composite number, the result may or may not...

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

all prime numbers n satisfy the above equation, as explained in the Euler's criterion article. The equation can be tested rather quickly, which can be used...

p − 1 2 = 1   ( mod   p ) a^{\frac {p-1}{2}}=1~({\text{mod}}~p) (Euler's Criterion) there is a square root of − 1 {\textstyle -1} in Z / p Z {\textstyle...

12(q−1)/2 − 1, same as 3(q−1)/2 ≡ 1 mod q and 12(q−1)/2 ≡ 1 mod q (per Euler's criterion) q − 3, q − 4, q − 9, q − 12 are quadratic nonresidues q − 3, q − 4...

{\displaystyle (10|13)} has to be equal to 1. This can be computed using Euler's criterion: ( 10 | 13 ) ≡ 10 6 ≡ 1 ( mod 13 ) . {\textstyle (10|13)\equiv 10^{6}\equiv...

provides needed factorization of f ( x ) {\displaystyle f(x)} . Due to Euler's criterion, for every monomial ( x − λ ) {\displaystyle (x-\lambda )} exactly...

64b2. For an odd prime number p and a quadratic residue a (mod p), Euler's criterion states that a p − 1 2 ≡ 1 ( mod p ) , {\displaystyle a^{\frac {p-1}{2}}\equiv...

In graph theory, Mac Lane's planarity criterion is a characterisation of planar graphs in terms of their cycle spaces, named after Saunders Mac Lane who...

In Itô calculus, the Euler–Maruyama method (also simply called the Euler method) is a method for the approximate numerical solution of a stochastic differential...

{1}{1-p^{-s}}}\cdots } Both sides of the Euler product formula converge for Re(s) > 1. The proof of Euler's identity uses only the formula for the geometric...