Fermat's Little Theorem Calculator
Verify Fermat's Little Theorem, find multiplicative inverses, and test primality
Calculate Fermat's Little Theorem
Any positive integer
Must be a prime number
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Fermat's Little Theorem
General Form:
a^p ≡ a (mod p)
For any integer a and prime p
Coprime Form:
a^(p-1) ≡ 1 (mod p)
When gcd(a,p) = 1
Examples
Example 1:
a = 3, p = 7
3^6 ≡ 1 (mod 7)
Since gcd(3,7) = 1
Example 2:
a = 14, p = 7
14^7 ≡ 14 (mod 7)
Since gcd(14,7) = 7 ≠ 1
Tips & Applications
p must be prime for the theorem to apply
Used in RSA cryptography
Helps find multiplicative inverses
Useful for primality testing
Foundation of modular arithmetic
Understanding Fermat's Little Theorem
What is Fermat's Little Theorem?
Fermat's Little Theorem is a fundamental result in number theory. It provides a relationship between modular exponentiation and prime numbers, forming the basis for many cryptographic algorithms.
The Two Forms:
General Form
For any integer a and prime p:
a^p ≡ a (mod p)
Coprime Form
When gcd(a,p) = 1:
a^(p-1) ≡ 1 (mod p)
Mathematical Applications
Cryptography
Foundation of RSA encryption and digital signatures
Multiplicative Inverses
Find a^(-1) ≡ a^(p-2) (mod p) when p is prime
Primality Testing
Fermat primality test (probabilistic)
Modular Arithmetic
Simplify large exponentiation problems
Historical Context
1640
Pierre de Fermat states the theorem in a letter to Frénicle de Bessy
1683
Leibniz discovers a proof but keeps it unpublished
1736
Euler publishes the first proof of Fermat's Little Theorem
Important Notes
• The theorem is called "little" to distinguish it from Fermat's Last Theorem
• The Fermat primality test can have false positives (Carmichael numbers)
• The theorem only works when p is prime
• When a and p are not coprime, use the general form a^p ≡ a (mod p)
• This theorem is fundamental to modern cryptography and computer science