Fermat’s little theorem

In this publication, we will consider one of the main theorems in the theory of integers –  Fermat’s little theoremnamed after the French mathematician Pierre de Fermat. We will also analyze an example of solving the problem to consolidate the presented material.

Statement of the theorem

1. Initial

If p is a prime number a is an integer that is not divisible by pthen a^p-1 – 1 divided by p.

It is formally written like this: a^p-1 ≡ 1 (against p).

Note: A prime number is a natural number that is only divisible by XNUMX and itself without remainder.

For example:

• a = 2
• p = 5
• a^p-1 – 1 = 2^{5 – 1} – 1 = 2⁴ – 1 = 16 – 1 = 15
• number 15 divided by 5 without a remainder.

2. Alternative

If p is a prime number, a any integer, then a^p comparable to a modulo p.

a^p ≡ a (against p)

History of finding evidence

Pierre de Fermat formulated the theorem in 1640, but did not prove it himself. Later, this was done by Gottfried Wilhelm Leibniz, a German philosopher, logician, mathematician, etc. It is believed that he already had the proof by 1683, although it was never published. It is noteworthy that Leibniz discovered the theorem himself, not knowing that it had already been formulated earlier.

The first proof of the theorem was published in 1736, and it belongs to the Swiss, German and mathematician and mechanic, Leonhard Euler. Fermat’s Little Theorem is a special case of Euler’s theorem.

Example of a problem

Find the remainder of a number 2¹² on 12.

Solution

Let’s imagine a number 2¹² as 2⋅2¹¹.

11 is a prime number, therefore, by Fermat’s little theorem we get:

2¹¹ ≡ 2 (against 11).

Hence, 2⋅2¹¹ ≡ 4 (against 11).

So the number 2¹² divided by 12 with a remainder equal to 4.