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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 .
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.
In 1736, Leonhard Euler published a proof of Fermat's little theorem [1] (stated by Fermat without proof), which is the restriction of Euler's theorem to the case where n is a prime number. Subsequently, Euler presented other proofs of the theorem, culminating with his paper of 1763, in which he proved a generalization to the case where n is ...
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 ...
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.
Euler's criterion Let p is an odd prime and a is an integer not divisible by p. Euler's criterion provides a slick way to determine whether a is a quadratic residue mod p. It says that is congruent to 1 mod p if a is a quadratic residue mod p and is congruent to -1 mod p if not. This can be written using Legendre symbols as
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.
The example for a=17 first of all tries to show how we can figure it out which numbers are squares modulo a prime without using Euler's criterion for some small a. Then we can watch: k 2 ≡ 17 (mod p). We get all p which solves it {2,4,8,16}. Now using Euler's criterion we see that: 17 (p-1)/2 ≡ 1 mod p; , for all p from {2,4,8,16}.