How can we find a large prime number
People use numbers whenever they do math. Yet, do they know that each number in the number system has its own unique trait? Numbers such as 4 and 9 are considered square numbers because 2 times 2 is 4, and 3 times 3 is 9. There also prime numbers. Prime numbers are numbers that have exactly two divisors. The number one is not included because it only has one divisor, itself. The smallest prime number is two, then three, then five, and so on. This list goes on forever and the largest known primes are called Mersenne primes. A Mersenne prime is written in the form of 2p-1. So far, the largest known Mersenne prime is 225,964,951-1, which is the 42nd Mersenne prime. This prime number has 7,816,230 digits!
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
2
wrong about the numbers 23, 29, 37, but was correct for the number 31....
... middle of paper ...
...a that can generate primes. Since there is an infinite amount of primes, we cannot conclude what the largest prime is. However, we do know that there are 25 primes less than 100, 168 less than 1000, 1229 less than 10,000, and as of January 2000, there
5
are 2,220,819,602,560,918,840 primes less than 100,000,000,000,000,000,000 (Flannery 69). However, by working with other people, perhaps we can use all of these methods to discover the next largest prime.
6
Works Cited
Caldwell, Chris K. Mersenne Primes: History, Theorems and Lists. 29 July 2005.
Flannery, Sarah. In Code. Chapel Hill: Algonquin Books of Chapel Hill, 2001.
Mersenne Prime Search. 06 March 2005. GIMPS. 27 July 2005.
Robert, A. Wayne and Dale E. Varberg. Faces of Mathematics. New York: Harper & Row Publishers, Inc., 1978.
Proposed in 1742 by Christian Goldbach, the Goldbach Conjectures have remained among the most famous mathematical problems ever proposed. The conjectures were first identified in a letter by him written to fellow famous mathematician, Leonhard Euler, the founder of Euler’s constant (2.718). The conjectures basically state that every even integer greater than two is a sum of two primes, and every odd integer greater than five is a sum of three primes. Though there is still a lot to discover about the conjectures, mathematicians generally view them as true. In this mathematical exploration I shall discuss in detail both of Goldbach’s conjectures and the mathematical procedures that occur.
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.
In very ancient times, 3 was used as the approximate value of pi, and not until Archimedes (3rd century BC) does there seem to have been a scientific effort to compute it; he reached a figure equivalent to about 3.14. A figure equivalent to 3.1416 dates from before AD 200. By the early 6th century Chinese and Indian mathematicians had independently confirmed or improved the number of decimal places. By the end of the 17th century in Europe, new methods of mathematical analysis provided various ways of calculating pi. Early in the 20th century the Indian mathematical genius Srinivasa Ramanujan developed ways of calculating pi that were so efficient that they have been incorporated into computer algorithms, permitting expressions of pi in millions of digits.
conceive of this number or anything that pertains to the infinite. There is always one more. A billion is a fairly large number, 1 with 9 zeros after it. If one counted by seconds without breaks, it would take over 32 years to reach it. A Google, is a number written as 1 with one hundred zeros after it. One couldn't even count the number of lifetimes it would take to count to this number. Yet there are even much higher numbers such as a Googleplex. This number is one with a Google zeros. It would take far far too long to even write out the number. If the entire known universe was packed with quarks, the smallest known material, the number of quarks would not add up to a Googleplex. Compared to infinity, though, this number is as far away as the number one.
[4] Nesterenko, Yu., V., A simple proof of the irrationality of π, Russ. J. Math. Phys. 13(2006),
As you can see, Euler provided a great deal to the world of mathematics. From developing notation, formulas, and important constants, to proving formulas and equations that stumped most other mathematicians of his day, there was almost nothing he could not do that involved mathematics. He was an instrumental figure in developing the future of modern mathematics and is credited in help developing pre-calculus, calculus, and differential equations. While he is not a household name, he is very easily the greatest mathematician to have ever lived.
Eventually the hype surrounding the conjecture died down in the early 1860’s. This down time, during which interest in the problem was minimal, only lasted about twenty years. A lawyer by the name of Alfred Bray Kempe proposed a solution in The American Journal of
Euclid wrote the most permanent mathematical works of all time, known as “The Element”. This comprised of 13 volumes of work filled with Mathematic theories and knowledge ranging from axioms, and...
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...
In conclusion, Fibonacci numbers are used throughout society. It is astonishing how these sets of never-ending numbers, are used in various ways. From being able to compute pi and being used in art, Fibonacci numbers are very unique compared to other mathematical subjects.
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.
Towers, J., Martin, L., & Pirie, S. (2000). Growing mathematical understanding: Layered observations. In M.L. Fernandez (Ed.), Proceedings of the Annual Meetings of North American Chapter of the International Group for the Psychology of Mathematics Education, Tucson, AZ, 225-230.
...ms that are present in today’s society would not be possible without Gauss’s effort on number theory.
Prime numbers have been of interest to mathematicians for centuries, and we owe much of our existing knowledge on the subject to thinkers who lived well before the Common Era––such as Euclid who demonstrated that there are infinitely many prime numbers around 300 BCE. Yet, for as long as primes have been an element of the mathematician’s lexicon, many questions about prime numbers remain unreso...
...re encompassing way, it becomes very clear that everything that we do or encounter in life can be in some way associated with math. Whether it be writing a paper, debating a controversial topic, playing Temple Run, buying Christmas presents, checking final grades on PeopleSoft, packing to go home, or cutting paper snowflakes to decorate the house, many of our daily activities encompass math. What has surprised me the most is that I do not feel that I have been seeking out these relationships between math and other areas of my life, rather the connections just seem more visible to me now that I have a greater appreciation and understanding for the subject. Math is necessary. Math is powerful. Math is important. Math is influential. Math is surprising. Math is found in unexpected places. Math is found in my worldview. Math is everywhere. Math is Beautiful.