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 (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...

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...

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...

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...

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...

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

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...

{\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...

{\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...

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

{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...

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...

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...

{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...