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.
and equimultiple n is taken of c and d, then a and b are in same ratio with c
Ball, Rouse. “Sir Isaac Newton.” A Short Account of the History of Mathematics. 4th ed. Print.
The Four Color Theorem was one of the first major theorem that was proved by the computer. It is the proof that can not be verified by many mathematicians. But the independent verification had convinced people that the theorem was finally proved. I believe because of the new technology, the proof of Four Colo Theorem will be improved in later time.
Both A and B’s answers appear to equate numeracy to math, which contradicts Australian curriculum’s definition, but, in a small way, fulfils the 21st century model’s (appendix 2) first requirement, that “a numerate person requires mathematical knowledge.” (Goos, 2014). Person A elaborates further
A Street Car Named Desire by Tennessee Williams is an iconic playwright throughout American Literature history. The play has many meanings to it, but the one meaning that stood out most and played an affect on the end of the play would be the treatment the two characters gave each other. The two characters are Blanche Dubois and Stanley Kowalski. Stanley’s treatment of Blanche throughout the play leads to her steady decline into madness.
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.
The mathematicians of Pythagoras's school (500 BC to 300 BC) were interested in numbers for their mystical and numerological properties. They understood the idea of primality and were interested in perfect and amicable numbers.
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.
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...
Calculus, the mathematical study of change, can be separated into two departments: differential calculus, and integral calculus. Both are concerned with infinite sequences and series to define a limit. In order to produce this study, inventors and innovators throughout history have been present and necessary. The ancient Greeks, Indians, and Enlightenment thinkers developed the basic elements of calculus by forming ideas and theories, but it was not until the late 17th century that the theories and concepts were being specified. Originally called infinitesimal calculus, meaning to create a solution for calculating objects smaller than any feasible measurement previously known through the use of symbolic manipulation of expressions. Generally accepted, Isaac Newton and Gottfried Leibniz were recognized as the two major inventors and innovators of calculus, but the controversy appeared when both wanted sole credit of the invention of calculus. This paper will display the typical reason of why Newton was the inventor of calculus and Leibniz was the innovator, while both contributed an immense amount of knowledge to the system.
As it can be seen the number for S is 12 which is a very high number to use as an exponent and can be difficult to calculate but using math we can figure it out. Using the formula ab(mod n) = ((a mod n)(b mod n)) mod n
Even the smallest tasks can impact the world in a significant way. Math, despite its trivial appearance, is large in grandeur that governs our world from the inside and the outside. The many twists and turns that exist in Mathematics make its versatility unparalleled and continues to awe the many Mathematicians today and the many more to come. The Binomial Theorem is one such phenomenon, which was founded by the combined efforts of Blaise Pascal, Isaac Newton and many others. This theorem is mainly algebraic, which contains binomial functions, arithmetic sequences and sigma notation. I chose the Binomial Theorem because of its complexity, yet simplicity. Its efficiency fascinates me and I would like to share this theorem that can be utilized to solve things in the Mathematical world that seem too daunting to be calculated by normal means.
Zeno of Elea was the next person who attempted to prove irrational numbers by challenging the Pythagorean mathematics as well. He lived from 490BC to 430BC. Zeno had influence from Socrate...
The 17th Century saw Napier, Briggs and others greatly extend the power of mathematics as a calculator science with his discovery of logarithms. Cavalieri made progress towards the calculus with his infinitesimal methods and Descartes added the power of algebraic methods to geometry. Euclid, who lived around 300 BC in Alexandria, first stated his five postulates in his book The Elements that forms the base for all of his later Abu Abd-Allah ibn Musa al’Khwarizmi, was born abo...
The abstractions can be anything from strings of numbers to geometric figures to sets of equations. In deriving, for instance, an expression for the change in the surface area of any regular solid as its volume approaches zero, mathematicians have no interest in any correspondence between geometric solids and physical objects in the real world. A central line of investigation in theoretical mathematics is identifying in each field of study a small set of basic ideas and rules from which all other interesting ideas and rules in that field can be logically deduced. Mathematicians are particularly pleased when previously unrelated parts of mathematics are found to be derivable from one another, or from some more general theory. Part of the sense of beauty that many people have perceived in mathematics lies not in finding the greatest richness or complexity but on the contrary, in finding the greatest economy and simplicity of representation and proof.