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.
INTRODUCTION OF FERMAT'S LITTLE THEOREM
Pierre de Fermat was a French mathematician whose contribution to analytic geometry and calculus are duly noted. But, what made him and still makes him a relevant mathematic figure is one of his well and widely known theorem; Fermat's Little Theorem. This theorem was first stated by him in a letter to a fellow friend on October 18, 1640, but what made it interesting is that he gave no proof of this theorem. This is how the theorem became well-known. It aroused mathematicians who came across this theorem to prove it. Though, this theorem has now been proved many times by many ways what still keeps it interesting is its real world application in the RSA security key system. In this exploration, I am going to focus mainly on the proofs of Fermat's Little Theorem.
Fermat's Little Theorem1 states that if 'p' is a prime number and 'a' is any integer, then,
ap-1 ≡ 1 (mod p) (1)
or
ap ≡ a (mod p) (2)
By multiplying equation ap-1 ≡ 1 (mod p) with 'a', we can get equation the equation ap ≡ a (mod p)
An example of how this theorem works is as follows:
In order to really understand how this theorem works, it is important to know some def...
... middle of paper ...
... system.
6
Bibliography
TURNAGE, CAROLINE LAROCHE. "SELECTED PROOFS OF FERMAT'S LITTLE THEOREM AND WILSON'S THEOREM." Thesis. WAKE FOREST UNIVERSITY, 2008. Sakai.wfu.edu. May 2008. Web. 27
Feb. 2014.
82ed8fda9d39/publications/Student/Caroline LaRoche Turnage - Thesis.pdf>.
"Fermat's Little Theorem." Wikipedia.org. N.p., n.d. Web. 2 Mar. 2014. .
"Proofs of Fermat's Little Theorem." En.wikipedia.org. N.p., July 2011. Web. 2 Mar. 2014. .
"RSA Public Key Encryption – The Code That Secures The internet." IB Maths Resources. N.p., n.d. Web. 2 Mar. 2014. .
Goldbach’s conjecture is one of the most well-known theories in all of mathematics. His conjecture states that, “every even integer greater than 2 can be expressed as the sum of two primes.” Goldbach’s conjecture includes the Goldbach number and many other algebraic expressions. Goldbach’s conjecture is so crucial that it was even featured in Hans Magnus Enzensberger’s The Number Devil. During the 5th night, the number devil shows Robert the Goldbach conjecture. On page 98 of The Number Devil, the number devil gives Robert examples of how to solve and work Goldbach’s conjecture. The number devil uses triangles as an example to introduce Goldbach’s conjecture. The number devil makes Robert throw coconuts to make triangles. This example shows a perfect example of Goldbach’s conjecture because it shows that “every even integer greater than 2 can be expressed as the sum of two primes.” The number
RSA encryption is the foundation of public key cryptography security products. For example, credit card companies use the RSA algorithm for customers’ individual online WebPages. The credit card companies publish a big number on WebPages, which is made by big prime numbers using the RSA algorithm. Since neither computers nor people can factor such big numbers, the RSA encryption system has secured many customers’ information.
The first proof, The Way of Motion, is about how things change in the world and how things are put into motion. Since you cannot infinitely regress backwards, there must be a first unmoved mover. This is understood to be God.
Many number theorists, who study certain properties of integers, have been trying to find formulas to generate primes. They believed that 2p-1 would always generate primes whenever p is prime. It turns out that if p is composite, then the number will also be a composite number. However, later mathematicians claimed that 2p-1 only works for certain primes p. For example, the number 11 is a prime because its divisors are only 1 and 11. In this case, 211-1 is 2047 and Hudalricus Regius showed that this number is composite in 1536 because 23 and 89 are factors of 2047. From then on, whenever a prime number can be written in the form of 2p-1, it is considered to be a Mersenne prime. Many conjectures have been made about p. Pietro Cataldi showed that 2p-1 was true for 17 and 19. However, he stated that it was also true for the prime numbers 23, 29, 31, and 37. Number theorists such as Fermat and Euler proved that Cataldi was
A prime number is an integer with only positive divisors one and itself. The ancient Greeks proved that there where infinitely many primes and that they where irregularly spaced. Mersenne examined prime numbers and wanted to discover a formula that would represent all primes. The formula is (2p-1) where p is a known prime number. Mersenne claimed that if a number n=(2p-1) is prime then p=2,3,5,7,13,17,31,67,127, and 257, but composite for the other forty-four primes smaller than or equal to 257. He was wrong about five primes less than or equal to 257. He claimed 67 and 257 had a p that was prime and he also missed three that did have a p that was prime. He would never be able to accomplish the task of creating a formula to represent all prime numbers; however the form he created is still used today when searching for large prime numbers.
“John Napier Discovers Logarithms” (2001) states that, “The primary reason for this is because his tables of logarithms vastly simplified computation” (para. 8). Logarithms have greatly helped mathematicians by speeding up calculations in the pre-calculator days. Mathematicians also found other uses for logarithms and invented other ways to apply them to problems. Although the most people in Napier’s time had no use for such invention, the discovery of logarithms has directly or indirectly affected almost everyone in some
My knowledge has grown over the past six years, outwith the areas of learning offered by school courses, and I see this course as an opportunity to gain new skills and broaden my knowledge further. My main interests are varied, including communications and the internet, system analysis and design, software development, processors and low level machine studies. I have recently developed an interest in data encryption, hence my active participation in the RSA RC64 Secret-Key challenge, the latest international de-encryption contest from the RSA laboratories of America.
For thousands of years cryptography and encryption have been used to secure communication. Military communication has been the leader of the use of cryptography and the advancements. From the start of the internet there has been a greater need for the use of cryptography. The computer had been invented in the late 1960s but there was not a widespread market for the use of computers really until the late 1980s, where the World Wide Web was invented in 1989. This new method of communication has called for a large need for information security. The internet allows people to communicate sensitive information, and if received into the wrong hands can cause many problems for that person.
By 1904 Ramanujan had begun to undertake deep research. He investigated the series (1/n) and calculated Euler's constant to 15 decimal places. He began to study the numbers, which is entirely his own independent discovery.
Fermat was born in 1601 in Beaumont-de-Lomagne, France and initially studied mathematics in Bordeaux with some of the disciples of Viete, a French algebraist (Katz 2009). He went on to earn a law degree and become a successful counselor. Mathematics was merely a hobby to him, so he never published because he did not want to thoroughly explain his discoveries in detail. He died in 1665 and his son later published his manuscripts and correspondence. Fermat adapted Viète’s algebra to the study of geometric loci and used letters to represent variable distances. He discovered that the study of loci, or sets of points with certain characteristics, could be made easier by applying algebra to geometry through a coordinate system (Katz 2009). Basically any relation between ...
Interesting and connection to higher level knowledge are significant elements for people to move on higher level academic fields. And the stories he used was emotional encouraged me to keep studying in mathematics by showing me the successful instances. Even I had to do a little research about his background to prove his credibility of talking about mathematics, it makes the editorial slightly less in credibility as a scientific publications. But after reading the editorial, I was deeply He was persuaded by Suri’s view about why recreational math is important, thereby, stimulated a strong interest in studying of recreational math. In conclusion, it is a convictive
Burton, D. (2011). The History of Mathematics: An Introduction. (Seventh Ed.) New York, NY. McGraw-Hill Companies, Inc.
The history of math has become an important study, from ancient to modern times it has been fundamental to advances in science, engineering, and philosophy. Mathematics started with counting. In Babylonia mathematics developed from 2000B.C. A place value notation system had evolved over a lengthy time with a number base of 60. Number problems were studied from at least 1700B.C. Systems of linear equations were studied in the context of solving number problems.
Throughout math, there are many patterns of numbers that have special and distinct properties. There are even numbers, primes, odd numbers, multiples of four, eight, seven, ten, etc. One important and strange pattern of numbers is the set of Fibonacci numbers. This is the sequence of numbers that follow in this pattern: 1, 1, 2, 3, 5, 8, 13, 21, etc. The idea is that each number is the sum of its previous two numbers (n=[n-1]+[n-2]) (Kreith). The Fibonacci numbers appear in various topics of math, such as Pascal?s Triangle and the Golden Ratio/Section. It falls under number theory, which is the study of whole or rational numbers. Number Theory develops theories, simple equations, and uses special tools to find specific numbers. Some topic examples from number theory are the Euclidean Algorithm, Fermat?s Little Theorem, and Prime Numbers.
...Morris & University of Georgia). That is the fundamental thought of the the theorem and can be expanded to solve more complex problems. Another contribution he made are the Pythagorean triples which are three positive integers that follow the a2 + b2 = c2 pattern (Wikipedia , 2013). When a triangle fits into this mold, they are referred to as Pythagorean Triangle (Wikipedia , 2013). Examples would be, 3,4,5 and 5,12,13. When those sets of numbers are seen it can be assumed that the triangle is a right triangle so you can go forth and using the Pythagorean Theorem to solve it. To get a triple set, you need to use Euclid’s formula (Wikipedia , 2013) .