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Fermat’s Little Theorem

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1. Introduction:
As I was looking for a theorem to prove for my Mathematics SL internal assessment, I couldn’t help but read about Fermat’s Little Theorem, a theorem I never heard of before. Looking into the theorem and reading about it made me develop an interest and genuine curiosity for this theorem. It was set forth in the 16th century by a French lawyer and amateur mathematician named Pierre de Fermat who is given credit for early developments that led to infinitesimal calculus. He made significant contributions to analytic geometry, probability, and optics. Fermat is best known for Fermat’s last theorem. Nevertheless, for the purpose of this investigation I will study his little theorem one of the beautiful proofs in Mathematics.

2. Fermat’s Little Theorem:
Fermat’s little theorem says that for a *prime number p and some natural number a, a p – a is divisible by p and will have a *remainder of 0.

*Prime number: A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. 3 is a prime number because only 1 and 3 evenly divide it.
*Remainder: The remainder is the number that is left over in a division in which one quantity does not exactly divide another. If we divide 23 by 3 the answer will be 7 and the remainder 2.

3. Examples:
Let’s say we have four natural numbers 2, 3, 4, 5 and the four prime numbers 2, 3, 5, 7 and we want to test Fermat’s little theorem:

When a = 2 and p = 2
22 – 2 = 2
2 ÷ 2 = 1 (clearly 2 is divisible by 2 and has a remainder of 0)

When a = 3 and p = 3
33 – 3 = 24
24 ÷ 3 = 8 (24 is divisible by 3 and has a remainder of 0)

When a = 4 and p = 5
45 – 4 = 1020
1020 ÷ 5 =204 (1020 is divisible by 5 and has a remainder of 0)

When a = 5 and p = 7
57 –...

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...is) as well as in number theory. The theorem is used in the encryption of data, which is the process of encoding information in such a way that only authorized parties can read it by unlocking the hidden information using a decryption key.
7. Conclusion
Proving the theorem myself truly gave me an understanding of its structure and many practical uses in Mathematics. Looking at it at first, I wouldn’t assume that it was such an important pieces in number theory. However, the theorems simplicity yet complex structure is what makes it useful in so many areas. Fermat’s Little Theorem is a theorem that I have never studied in class nor have I the change to work with. It is a very new concept to my knowledge that has definitely enriched my Mathematical view. I am eager to learn more about Euler and his other theorems. I am particularly interested in proving his theorems.
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