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Math Forum: Ask Dr. Math FAQ: Prime Numbers    Ask Dr. Math: FAQ Prime Numbers   Dr. Math FAQ || Classic Problems || Formulas || Search Dr. Math || Dr. Math Home What is a prime number? How can you find prime numbers? What's the 'Sieve of Eratosthenes'? How can you decide if a number is prime? What's the largest known prime?A prime number is a positive integer that has exactly two positiveinteger factors, 1 and itself. For example, if we list the factors of28, we have 1, 2, 4, 7, 14, and 28. That's six factors. If we listthe factors of 29, we only have 1 and 29. That's two factors. So wesay that 29 is a prime number, but 28 isn't. Another way of saying this is that a prime number is a positiveinteger that is not the product of two smaller positive integers.Note that the definition of a prime number doesn't allow 1 to be aprime number: 1 only has one factor, namely 1. Prime numbers haveexactly two factors, not "at most two" or anything likethat. When a number has more than two factors it is called a compositenumber.Here are the first few prime numbers:2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, etc.The Sieve of EratosthenesEratosthenes (275-194 B.C., Greece) devised a 'sieve' to discoverprime numbers. A sieve is like a strainer that you use to drain spaghettiwhen it is done cooking. The water drains out, leaving yourspaghetti behind. Eratosthenes's sieve drains out composite numbersand leaves prime numbers behind. To use the sieve of Eratosthenes to find the prime numbers up to 100,make a chart of the first one hundred positive integers (1-100): 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100Cross out 1, because it is not prime.Circle 2, because it is the smallest positive even prime. Now cross out every multiple of 2; in other words, cross out every second number. Circle 3, the next prime. Then cross out all of the multiples of 3; in other words, every third number. Some, like 6, may have already been crossed out because they are multiples of 2.Circle the next open number, 5. Now cross out all of the multiples of 5, or every 5th number.Continue doing this until all the numbers through 100 have either been circled or crossed out. You have just circled all the prime numbers from 1 to 100!There are various primality tests, from very simple to very complex, which allow you to determine if a given number is prime. You can read more about them atPrimality Testing in our Selected Answers.There is no largest prime number, but the effort to find ever-larger primes is ongoing and you can read about The Largest Known Primes on the Web. From the Dr. Math archives:See Middle School Prime Numbers or search the Dr. Math archives using the words "prime number" (that exact phrase; just the words, not the quotes) to find questions and answers about prime numbers at all levels. Here are some good places to start:Finding Prime NumbersFinding Prime Numbers (2)Frequency of PrimesIs Zero Prime or Composite?Large Prime NumbersNo Largest Prime Numberp, p+8, p+22 Not PrimePrimality TestPrime FactorizationPrime Number FormulaPrime Number InformationPrime Number TheoremsPrime Numbers 20-30Relative PrimesWhy is 1 Not Considered Prime?Writing a Program to Generate Primes On the Web:Finding Primes and Proving PrimalityThe First 10,000 PrimesThe Great Internet Mersenne Prime SearchHow Many Primes Are There?The Largest Known PrimesMersenne Prime - Susan StepneyMersenne Primes - Jon VinopalMersenne Primes: History, Theorems and ListsPrime Number (Eric Weisstein's World of Mathematics)Prime Numbers and FactorsThe Prime Pages - Chris Caldwell  73939133 - Prime Numbers - Amazing Number FactsPrime Theorem of the CenturyStudents' Mersenne Prime PageWhy do people find these primes? - Chris CaldwellSubmit your ownquestion to Dr. Math[Privacy Policy] [Terms of Use]Math Forum Home ||Math Library ||Quick Reference ||Math Forum SearchAsk Dr. Math ®© 1994-2008 Drexel University. All rights reserved.http://mathforum.org/ The Math Forum is a research and educational enterprise of the Drexel School of Education. |
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