🔢 Fermat's Little Theorem Calculator
Verify Fermat's Little Theorem for modular arithmetic
How to Use This Calculator
Enter Base and Prime
Enter a (base) and p (prime modulus). They must be coprime.
Click Verify
Press button to check if a^(p-1) ≡ 1 (mod p).
Fermat's Little Theorem
If p is prime and gcd(a, p) = 1, then:
a^(p-1) ≡ 1 (mod p)
Example: a = 3, p = 7
Check: 3^(7-1) = 3^6 ≡ ? (mod 7)
3^1 = 3 mod 7 = 3
3^2 = 9 mod 7 = 2
3^6 = (3^2)^3 = 2^3 = 8 ≡ 1 (mod 7) ✓
About Fermat's Little Theorem Calculator
Fermat's Little Theorem states that if p is prime and a is coprime to p, then a raised to the (p-1) power is congruent to 1 modulo p.
When to Use This Calculator
- Number Theory: Verify modular exponentiation
- Cryptography: Understand RSA and modular arithmetic
- Education: Learn Fermat's theorem
- Research: Check number theory properties
Why Use Our Calculator?
- ✅ Instant Verification: Check theorem instantly
- ✅ Modular Exponentiation: Efficient calculation
- ✅ Educational: Learn theorem concepts
- ✅ Completely Free: No registration required
Requirements
- p must be a prime number
- a and p must be coprime (GCD = 1)
- Theorem states: a^(p-1) ≡ 1 (mod p)
Frequently Asked Questions
What is Fermat's Little Theorem?
If p is prime and gcd(a,p)=1, then a^(p-1) ≡ 1 (mod p). A fundamental result in number theory.
Why must a and p be coprime?
If a and p share a factor, the theorem doesn't hold. For example, 2^(4-1) mod 4 = 8 mod 4 = 0 ≠ 1 (2 and 4 not coprime).
What is this used for?
Prime testing (Miller-Rabin), modular inverses, and cryptographic algorithms like RSA rely on this theorem.