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...

Eisenstein's criterion, in mathematics, gives sufficient conditions for a polynomial to be irreducible over the rational numbers Euler's criterion, in number...

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...

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...

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...

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...

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...

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

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...

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...

{\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...

{\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...

then p must divide at least one of a or b. Euler's criterion Euler's criterion Euler's theorem Euler's theorem states that if n and a are coprime positive...

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...

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

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...

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...

{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...

{\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...

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

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...

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, q-9, and...

semi-implicit Euler method, also called symplectic Euler, semi-explicit Euler, Euler–Cromer, and Newton–Størmer–Verlet (NSV), is a modification of the Euler method...

n + 1 ) / 2 {\displaystyle Q^{(n+1)/2}} . If n is prime, then, by Euler's criterion, Q ( n − 1 ) / 2 ≡ ( Q n ) ( mod n ) {\displaystyle Q^{(n-1)/2}\equiv...

prime. Necessity: assume that F n {\displaystyle F_{n}} is prime. By Euler's criterion, 3 ( F n − 1 ) / 2 ≡ ( 3 F n ) ( mod F n ) {\displaystyle 3^{(F_{n}-1)/2}\equiv...

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...

{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...

interpretation of the criterion, but not the substance of it, some texts continue to refer to the criterion as simply the 'Coulomb criterion'. The Mohr–Coulomb...