Prime numbers and their properties were first studied extensively by the ancient Greek mathematicians.
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.
A perfect number is one whose proper divisors sum to the number itself. e.g. The number 6 has proper divisors 1, 2 and 3 and 1 + 2 + 3 = 6, 28 has divisors 1, 2, 4, 7 and 14 and 1 + 2 + 4 + 7 + 14 = 28.
A pair of amicable numbers is a pair like 220 and 284 such that the proper divisors of one number sum to the other and vice versa.
You can see more about these numbers in the History topics article Perfect numbers.
By the time Euclid's Elements appeared in about 300 BC, several important results about primes had been proved. In Book IX of the Elements, Euclid proves that there are infinitely many prime numbers. This is one of the first proofs known which uses the method of contradiction to establish a result. Euclid also gives a proof of the Fundamental Theorem of Arithmetic: Every integer can be written as a product of primes in an essentially unique way.
Euclid also showed that if the number 2n - 1 is prime then the number 2n-1(2n - 1) is a perfect number. The mathematician Euler (much later in 1747) was able to show that all even perfect numbers are of this form. It is not known to this day whether there are any odd perfect numbers.
In about 200 BC the Greek Eratosthenes devised an algorithm for calculating primes called the Sieve of Eratosthenes.
There is then a long gap in the history of prime numbers during what is usually called the Dark Ages.
... middle of paper ...
If p is a prime, is 2p - 1 always square free? i.e. not divisible by the square of a prime.
Does the Fibonacci sequence contain an infinite number of primes?
Here are the latest prime records that we know.
The largest known prime (found by GIMPS [Great Internet Mersenne Prime Search] in December 2001) is the 39th Mersenne prime: M13466917 which has 4053946 decimal digits.
The largest known twin primes are 242206083 238880 1. They have 11713 digits and were announced by Indlekofer and Ja'rai in November, 1995.
The largest known factorial prime (prime of the form n! 1) is 3610! - 1. It is a number of 11277 digits and was announced by Caldwell in 1993.
The largest known primorial prime (prime of the form n# 1 where n# is the product of all primes n) is 24029# + 1. It is a number of 10387 digits and was announced by Caldwell in 1993.
Need Writing Help?
Get feedback on grammar, clarity, concision and logic instantly.Check your paper »
- ... Jane is exhibited to have qualities of forgiveness and understanding of conditions as she is empathetic. Jane is treated with disrespect because of Mrs. Reeds antagonizing feelings towards her, and this leads to her being ill-treated. A poor family structure can lead to an individual becoming self-destructive and can pertain long term effects that have a significant impact on the psychological state of the individual that determines not only their personality but their ability to make wise decisions.... [tags: The Solitude of Prime Numbers, Jane Eyre]
1790 words (5.1 pages)
- ... But nobody knows who first noticed and started telling people about happy numbers. But they never really left the classroom until Reg Allenby found out about them and started to show people what they were. Happy numbers Happy numbers are numbers that when you square the digits of the number and add the squares together repeatedly, in the end the answer will be one. This is how you can find a happy number: 19: 1²+9²= 1+81= 82 8²+2²= 64+4= 68 6²=8²= 36+64= 100 1²+0²+0²= 1+0+0= 1 As you can see 19 is a happy number because it equals 1 after having squared the digits a couple of times.... [tags: squaring, russia, prime]
658 words (1.9 pages)
- Theory of Numbers Mersenne Marin Mersenne was a French number theorist who lived from 1588 to 1648. Mersenne attended the College of Mans, the Jesuit College, and then Sorbonne to study theology. In 1611, he joined the religious order of the Minims. Once in the order, Mersenne continued his studies at Nigeon and Meaux. He became a priest at the Place Royale. The area in Paris, where Mersenne taught, became a meeting ground for Fermat, Pascal, and others who later became the core of the French Academy.... [tags: Biography, History, Mathematics]
325 words (0.9 pages)
- 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.... [tags: Mathematics Math]
1336 words (3.8 pages)
- 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.... [tags: prime number, mathematics]
864 words (2.5 pages)
- 1. World War II was a prime time for the development of computers of all kinds for many countries. This “war brought a host of computational problems which had to be solved in a hurry” in order to hope to win the war. (Lavington) Some areas for these new computing devices were designed for flight simulation, gunnery control, and aircrew training.” This period in time sparked a lot of creation and need for computers as many thought they would help win the war, and was essential to the development of computers.... [tags: Computer, ENIAC, Konrad Zuse]
1174 words (3.4 pages)
- Letter to Prime Minister Regarding the Homeless Mr Tony Blair, Prime minister, The House of Commons, London, Dear Mr Blair, I am writing in order to bring to your attention the harrowing numbers of young homeless people being practically pushed out onto the life on the streets. In my opinion with your help I feel that the figures can and will be dramatically decreased and better help will be provided all around. Im sure that somebody in your position would not be oblivious to the rough numbers of homeless people on the streets, but I also understand that facts and figures must be part of your daily routine so it cant be as easy to remember in de... [tags: Papers]
654 words (1.9 pages)
- Charlie Brooker’s Netflix series Black Mirror’s season one episode The National Anthem discusses how technology can be cruel and enabling. Also, the show represents how media thrives on the negative of the world. In the article titled “Scary Numbers” author Joel Best states, “Amid a cacophony of competing claims, advocates must make the case that their particular problem merits concern.” Also, “Advocates seeking to raise concern naturally find it advantageous to accentuate the negative; therefore, they prefer scary statistics that portray the problem as very common or very serious.”1 This quote basically summons up the entire episode in only a few lines.... [tags: Prime minister, Cabinet, Westminster system]
1920 words (5.5 pages)
- Set Theory in the Flesh The idea of infinity has been around for thousands of years. It it impossible to even 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.... [tags: Numbers Mathematics Essays]
1799 words (5.1 pages)
- Recurring Decimals Infinite yet rational, recurring decimals are a different breed of numbers. Mathematicians, in turn, have been fascinated by these special numbers for over two thousand years. The Hindu-Arabic base 10 system we use today was inspired by the Chinese method of decimals which was actually around 10000 years old. Decimals may have been around for a very long time, but what about recurring decimals. In fact the ancient Greeks were one of the first to deal with recurring decimals.... [tags: Mathematics Math Infinite Numbers]
1400 words (4 pages)