GIMPS harnesses the power of hundreds of thousands of small computers like yours to search for these "needles in a haystack".

Most GIMPS members join the search for the thrill of possibly discovering a record-setting, rare, and historic new Mersenne prime. The search for more Mersenne primes is already under way. There may be smaller, as yet undiscovered Mersenne primes, and there are certainly larger Mersenne primes waiting to be discovered. Entropia's enterprise distributed and grid computing products harness the vast untapped processing power of PCs on corporate networks to build virtual computing clusters of a selected design and scope.

The technology, scalable to millions of PCs, delivers computing capacities on par with that of supercomputers at a fraction of the cost. Companies deploy Entropia applications to accelerate their time to market using the PCs they already own, dramatically increasing the ROI in their desktop computing and network assets. Commercial applications include critical pharmaceutical, chemical and materials research, and financial services.

## Prime number

Entropia application grids power mathematics research www. For more information visit the Entropia Web site at www. In June , Nayan Hajratwala discovered the previous largest known prime number in the United States. Joel Armengaud discovered the 35th Mersenne prime in November, , in France. Prime numbers have long fascinated amateur and professional mathematicians. An integer greater than one is called a prime number if its only divisors are one and itself. The first prime numbers are 2, 3, 5, 7, 11, etc. For example, the number 10 is not prime because it is divisible by 2 and 5.

A Mersenne prime is a prime of the form 2 p The first Mersenne primes are 3, 7, 31, , etc.

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The evolutionary strategy used by cicadas of the genus Magicicada makes use of prime numbers. They only pupate and then emerge from their burrows after 7, 13 or 17 years, at which point they fly about, breed, and then die after a few weeks at most. Biologists theorize that these prime-numbered breeding cycle lengths have evolved in order to prevent predators from synchronizing with these cycles. Prime numbers have influenced many artists and writers.

The French composer Olivier Messiaen used prime numbers to create ametrical music through "natural phenomena". According to Messiaen this way of composing was "inspired by the movements of nature, movements of free and unequal durations". In his science fiction novel Contact , scientist Carl Sagan suggested that prime factorization could be used as a means of establishing two-dimensional image planes in communications with aliens, an idea that he had first developed informally with American astronomer Frank Drake in From Wikipedia, the free encyclopedia.

For other uses, see Prime disambiguation. Integer greater than 1 that has no positive integer divisors other than itself and 1. Main article: List of prime numbers. Main article: Fundamental theorem of arithmetic. Main article: Euclid's theorem. Main article: Formula for primes. Further information: Category:Conjectures about prime numbers.

Main articles: Prime number theorem and Prime-counting function. Main article: Riemann hypothesis. Main article: Modular arithmetic. Main article: p-adic number. Main articles: Prime element and Irreducible element. Main article: Prime ideals. Main article: Trial division.

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The sieve of Eratosthenes starts with all numbers unmarked gray. It repeatedly finds the first unmarked number, marks it as prime dark colors and marks its square and all later multiples as composite lighter colors. After marking the multiples of 2 red , 3 green , 5 blue , and 7 yellow , all primes up to the square root of the table size have been processed, and all remaining unmarked numbers 11, 13, etc. Main article: Sieve of Eratosthenes. Further information: List of prime numbers.

Main article: Integer factorization. Oxford University Press. Dyslexia, Dyscalculia and Mathematics: A practical guide 2nd ed. Golden Press. Barron's Educational Series. Elementary number theory 2nd ed. Freeman and Co. Elementary Theory of Numbers.

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North-Holland Mathematical Library. Notices of the American Mathematical Society. Numbers and Geometry. Undergraduate Texts in Mathematics.

A Selection of Problems in the Theory of Numbers. New York: Macmillan. Elementary Methods in Number Theory. Graduate Texts in Mathematics. How did the ancient Egyptian scribe prepare it?

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Archive for History of Exact Sciences. Mathematics and Its History. Undergraduate Texts in Mathematics 3rd ed. Scientific American. Bibcode : SciAm.

Mathematics Magazine. Fermat's Little Theorem November , p. Edward How Euler Did Even More. Mathematical Association of America. Elementary Number Theory with Applications.

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Academic Press. Goldbach Conjecture. Series In Pure Mathematics. World Scientific. Springer Monographs in Mathematics. In Bambah, R. Number Theory. Trends in Mathematics. Dirichlet's Theorem on Primes in Arithmetical Progressions". Introduction to Analytic Number Theory. New York; Heidelberg: Springer-Verlag.

Proth's Primality Test". Elementary Number Theory and Its Applications 4th ed. Barry; Hodges, Andrew The Once and Future Turing. Cambridge University Press.

Science in Context. Elementary Number Theory. Textbooks in mathematics. CRC Press. Secret History: The Story of Cryptology.

## Computational Number Theory

Discrete Mathematics and Its Applications. Dolciani mathematical expositions. Journal of Integer Sequences. For a selection of quotes from and about the ancient Greek positions on this issue, see in particular pp. For the Islamic mathematicians, see p. See in particular the entries for Stevin, Brancker, Wallis, and Prestet. Prime Numbers and Computer Methods for Factorization 2nd ed. The Book of Numbers. New York: Copernicus. For the sum of divisors, see Sandifer, C.