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In number theory, Euler's criterion is a formula for determining whether an integer is a quadratic residue modulo a prime. Precisely, Precisely, Let p be an odd prime and a be an integer coprime to p .
by Euler's criterion, but both sides of this congruence are numbers of the form , so they must be equal. Whether 2 {\displaystyle 2} is a quadratic residue can be concluded if we know the number of solutions of the equation x 2 + y 2 = 2 {\displaystyle x^{2}+y^{2}=2} with x , y ∈ Z p , {\displaystyle x,y\in \mathbb {Z} _{p},} which can be ...
By Euler's criterion, which had been discovered earlier and was known to Legendre, these two definitions are equivalent. [2] Thus Legendre's contribution lay in introducing a convenient notation that recorded quadratic residuosity of a mod p.
In mathematics and computational science, the Euler method (also called the forward Euler method) is a first-order numerical procedure for solving ordinary differential equations (ODEs) with a given initial value.
Modulo 2, every integer is a quadratic residue. Modulo an odd prime number p there 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 work within the multiplicative group of nonzero elements of the field (/).
The idea for the general proof follows the above supplemental case: Find an algebraic integer that somehow encodes the Legendre symbols for p, then find a relationship between Legendre symbols by computing the qth power of this algebraic integer modulo q in two different ways, one using Euler's criterion the other using the binomial theorem.
If the Euler's criterion formula is used modulo a composite number, the result may or may not be the value of the Jacobi symbol, and in fact may not even be −1 or 1. For example, For example, ( 19 45 ) = 1 and 19 45 − 1 2 ≡ 1 ( mod 45 ) .
If n is an odd composite integer that satisfies the above congruence, then n is called an Euler–Jacobi pseudoprime (or, more commonly, an Euler pseudoprime) to base a. As long as a is not a multiple of n (usually 2 ≤ a < n ), then if a and n are not coprime, n is definitely composite, as 1 < gcd ( a , n ) < n is a factor of n .