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  • Wilson's Theorem
  • Euler's Theorem
  • Fermat's Little Theorem
  • Reference

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  1. Fundamentals
  2. Modular Arithmetic

Theorems of Wilson, Euler, and Fermat

PreviousModular ArithmeticNextFermat's Little Theorem in Detail

Last updated 3 years ago

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Wilson's Theorem

A positive integer n>1n > 1n>1is a prime if and only if:

(n−1)!≡−1mod  n(n-1)! \equiv -1 \mod n (n−1)!≡−1modn

Euler's Theorem

Let n∈Z+n \in \mathbb{Z}^{+}n∈Z+ and a∈Za \in \mathbb{Z}a∈Z s.t. gcd(a,n)=1gcd(a, n) = 1gcd(a,n)=1, then:

aϕ(n)≡1mod  na^{\phi(n)} \equiv 1 \mod naϕ(n)≡1modn

Fermat's Little Theorem

Let pppbe a prime and a∈Za \in \mathbb{Z}a∈Z, then:

ap≡amod  pa^p \equiv a \mod pap≡amodp

or equivalently:

Reference

ap−1≡1mod  pa^{p-1} \equiv 1 \mod pap−1≡1modp

Wilson's Theorem - Brilliant
Euler's Theorem - Brilliant
Fermat's Little Theorem - Brilliant