Fermat's Little Theorem Calculator

Fermat's little theorem is one of the fundamental results of number theory. It says that if p is a prime number and a is an integer, then a^p – a is divisible by p. In modulo notation:

a^p ≡ a (mod p)

In particular, if a is not divisible by p, then:

a^p-1 ≡ 1 (mod p)

🙋 The condition that a is not divisible by p can be equivalently stated as a and p being coprime (or relatively prime) numbers. The quickest way to verify this requirement is to compute their GCF. See our relatively prime calculator to learn more.

Once we know what Fermat's little theorem is, let's see how it works in practice. We'll go together through some examples of Fermat's little theorem applied to concrete numbers.

Example 1

Let's consider p = 7 and a = 15. a is not divisible by p, so the theorem says that

15^7-1 = 15⁶ ≡ 1 (mod 7)

Indeed, we have

15⁶ – 1 = 11390624 = 7 × 1627232

Example 2

Be careful to check the assumptions! If p = 7 and a = 14, then 14⁶ is not equal to 1 modulo 7:

14⁶ – 1 ≡ 6 (mod 7)

This is because 14 and 7 are not coprime — 14 is divisible by 7. So, we should use the version of Fermat's little theorem for non-coprime numbers — the one that states a^p ≡ a (mod p):

14⁷ ≡ 14 (mod 7)

Indeed:

14⁷ – 14 = 7 × 15059070

🔎 Why is Fermat's little theorem called "little"? It's to distinguish it from Fermat's "great" or "last" theorem, which says that if n is an integer greater than 2, then no three positive integers x, y, and z satisfy the equation xⁿ + yⁿ = zⁿ. This theorem was stated by Fermat in the 17th century but proved only in 1995 by Andrew Wiles.

calculator for Fermat's little theorem is really simple to use:

1. Our tool accepts two inputs: the integers a and p.

2. Remember that p must be a prime number. You can use prime number calculator to verify if the number you want to enter is prime. If it is not, our Fermat's little theorem calculator will display a warning and refuse to cooperate further.

3. Our calculator will also check if your numbers are coprime to decide which version of Fermat's last theorem is appropriate.

4. In either case, the statement of Fermat's little theorem applied to your input will appear.

To find the multiplicative inverse of a modulo p using Fermat's little theorem:

1. Write the claim of Fermat's little theorem as a^p-1 ≡ 1 (mod p).
2. Recall the assumptions: p must be prime and a must not be a multiple of p.
3. Verify that your numbers satisfy these assumptions.
4. Rewrite the claim a × a^p-2 ≡ 1 (mod p).
5. That is, the multiplicative inverse of a modulo p is equal to a^p-2.

To check if p is a prime number using Fermat's little theorem:

1. Pick a random integer a ∈ {2, ... p-2} not divisible by p.
2. Compute a^p-1 (mod p).
3. If the result is not 1, then p is definitely not prime.
4. If the result is equal to 1, then we choose a different a and repeat the procedure from Step 2.
5. If the test cannot decide after many repetitions, then p is probably prime.