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 –...

... middle of paper ...

...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.

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 –...

... middle of paper ...

...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.

Related

- Better Essays
## Mathematical Exploration: Exploring the Proofs of Fermat's Little Theorem

- 1381 Words
- 3 Pages
- 5 Works Cited

Unraveling the complex and diverse nature of numbers has always been a fascinating ordeal for me; that is what makes and keeps me interested in the world of mathematics. Finding out new number patterns and the relationship between numbers is nothing short of a new discovery; that is how my interest into learning more about and exploring Fermat's Little Theorem came about.

- 1381 Words
- 3 Pages
- 5 Works Cited

Better Essays - Good Essays
## Pierre De Fermat

- 858 Words
- 2 Pages
- 1 Works Cited

Pierre de Fermat Pierre de Fermat was born in the year 1601 in Beaumont-de-Lomages, France. Mr. Fermat's education began in 1631. He was home schooled. Mr. Fermat was a single man through his life. Pierre de Fermat, like many mathematicians of the early 17th century, found solutions to the four major problems that created a form of math called calculus. Before Sir Isaac Newton was even born, Fermat found a method for finding the tangent to a curve. He tried different ways in math to improve the system. This was his occupation. Mr. Fermat was a good scholar, and amused himself by restoring the work of Apollonius on plane loci. Mr. Fermat published only a few papers in his lifetime and gave no systematic exposition of his methods. He had a habit of scribbling notes in the margins of books or in letters rather than publishing them. He was modest because he thought if he published his theorems the people would not believe them. He did not seem to have the intention to publish his papers. It is probable that he revised his notes as the occasion required. His published works represent the final form of his research, and therefore cannot be dated earlier than 1660. Mr. Pierre de Fermat discovered many things in his lifetime. Some things that he did include: -If p is a prime and a is a prime to p then ap-1-1 is divisible by p, that is, ap-1-1=0 (mod p). The proof of this, first given by Euler, was known quite well. A more general theorem is that a0-(n)-1=0 (mod n), where a is prime...

- 858 Words
- 2 Pages
- 1 Works Cited

Good Essays - Good Essays
## Bernoulli Essay

- 1722 Words
- 4 Pages

Born in the Netherlands, Daniel Bernoulli was one of the most well-known Bernoulli mathematicians. He contributed plenty to mathematics and advanced it, ahead of its time. His father, Johann, made him study medicine at first, as there was little money in mathematics, but eventually, Johann gave in and tutored Daniel in mathematics. Johann treated his son’s desire to lea...

- 1722 Words
- 4 Pages

Good Essays - Good Essays
## Integer Constant Essay

- 850 Words
- 2 Pages

Integer Constant An integer constant is made up of digits without decimal point. Rules The integer constant is formed with digits 0 to 9 Commas and blank spaces are not allowed.

- 850 Words
- 2 Pages

Good Essays - Good Essays
## Kent Bessey's Ethos Chapter 3

- 524 Words
- 2 Pages

In Kent Bessey’s book, I found that by using this principle, amazing conclusions can be drawn that would be impossible, or difficult but, on the other hand turn out to be true. I wasn’t sure why this type of information was

- 524 Words
- 2 Pages

Good Essays - Best Essays
## The Foundation of The United States of America

- 1795 Words
- 4 Pages
- 12 Works Cited

“The great book of nature can be ready only by those who know the language in which it was written. And that language is mathematics.”(Galilei, Galileo). Math is all around us, when people think of math, they think of equation and calculation. However, math is arguably one of most crucial and fundamental elements that controls and keeps our live going. Athens Greece is the foundation of math, with the brilliant mind of Archimedes and Pythagoras emerge the ideas of Pi and the Pythagorean Theorem. These two simple concepts helped United States and the world to achieve ground breaking and revolutionary discoveries. Pythagorean Theorem can be simply used in the architectural sense, but it can also be used to pinpoint two reference points. This simple equation is what modern GPS system uses to pinpoint and calculate the designated route, NASA uses advancement mathematics and Pythagorean theorem to determine the ...

- 1795 Words
- 4 Pages
- 12 Works Cited

Best Essays - Satisfactory Essays
## RSA encryption

- 1156 Words
- 3 Pages

RSA encryption All over the world there are millions of people use credit card and on-line shopping. Every individual gets different numbers for credit card and for transcription of on-line-shopping. Where did all this number come from? Are the numbers in order? No, those numbers are made by RSA algorithm.

- 1156 Words
- 3 Pages

Satisfactory Essays - Good Essays
## IMP 2 POW 8

- 1142 Words
- 3 Pages

Problem Statement My task was to find 3 equations, that would give me an answer, if I had certain information. The first was to find one that if you knew that there were four pegs on the boundary, and none on the interior, you could get the area. The second was if you knew that there were 4 pegs on the boundary, and you knew how many were on the interior, you could get the area. And last, if you had the number on the interior, and the number on the boundary, you could get the area.

- 1142 Words
- 3 Pages

Good Essays - Better Essays
## History Of Accounting

- 1195 Words
- 3 Pages

Once upon a time, Luca Pacioli wrote a math book. It was just a little survey and should have been treated like ordinary books of the time and read and then disappeared into historical archives and forgotten. A few brief chapters on practical mathematics made this one special.

- 1195 Words
- 3 Pages

Better Essays - Powerful Essays
## Encryption Method: The Decryption Method

- 2953 Words
- 6 Pages

Rivest, R.L., et al [1977]: An encryption method was presented in the paper with the then relatively new and novel property that publicly revealing an encryption key did not thereby reveal the corresponding decryption key. This had two important consequences: 1) Couriers or other secure means were not needed to transmit keys, since a message could be enciphered using an encryption key publicly revealed by the intended recipient. Only he could decipher the message, since only he knew the corresponding decryption key. 2) A message could be “signed" using a privately held decryption key. Anyone could verify this signature using the corresponding publicly revealed encryption key. Signatures could not be forged, and a signer could not later deny the validity of his signature. This had obvious applications in “electronic mail" and “electronic funds transfer" systems. Their encryption function was the only candidate for a “trap-door one-way permutation" known to the authors. However, they conceded that it might be desirable to find other examples, to provide alternative implementations should the security of their system turn out someday to be inadequate. This technique though, remains the most popular technique still in use. [1]

- 2953 Words
- 6 Pages

Powerful Essays