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Title: Math/Number Theory/Prime Numbers - Some Prime Numbers Sieves and factoring for small numbers. Download primes below 8 billion.
Status_of_Search_for_Multifactorial_Primes An organized search for primes of the form (n)(n-k)(n-2k)...+/-1. Includes table of numbers searched and primes found.

Thomas_R__Nicely Includes twin prime counts, tables of first occurrence prime gaps, papers.

The_Time_Traveler A group with new propositions in various areas of number theory.

Visualizing_the_Distribution_of_Prime_Numbers Investigation into patterns in the distribution of the primes by visualizing them.

Wikipedia__Prime_Numbers Index to many prime-related articles.

World_of_Numbers__World_of_Palindromic_Primes Records, statistics, curios and puzzles about primes reading the same backwards. Compiled by Patrick De Geest.


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Some Prime Numbers

Curious sheep in Ireland looking at the photographerThis page is sponsored by http://www.A-Wee-Bit-of-Ireland.com/.™Your source in the USA for the finest Irish wool sweaters, ruanas,capes, scarves, perfumes,caps and hats.All products made in Ireland.If you want many primes or larger primes, my JAVA prime number program might be better than this programbecause it will run on your computer instead of on my slow server.Or, you can download programs or source from my home page: http://www.rsok.com/~jrm/.If you only want thefirst 100 primes,they are athttp://www.rsok.com/~jrm/first100primes.html.http://www.rsok.com/~jrm/next_ten_primes.html will print the next 10primes after the integer you supply. If the graphics window is too small tohold the results, please check the JAVA console window.A sieve was used to generate a list of prime numbers. In order to reduce storage requirements, the information about prime or notprime was stored for 30 integers in each byte. Only one bit is neededto store prime or not prime for an integer. The value of the integeris known by the location of the bit.In each 30 integers, for N>= 1, the numbers that might be prime areN*30+1,N*30+7,N*30+11,N*30+13,N*30+17,N*30+19,N*30+23,N*30+29This means that information about prime or not prime only needs to occupy 8 bits. A ten byte file would hold prime or not prime for 300integers. It works because N*30 has 2,3,5 as factors, no matter whatthe value of N. (N*30)+3 has 3 as a factor since 3 is also a factorof 30. (N*30)+(x*3) also has 3 as a factor since it could be writtenas (N*10 + x)*3.The relationship used for this compact storage of prime numbers mightbe described by:(2-1)*(3-1)*(5-1)----------------- == 8 bits / 30 integers 2*3*5In a traditional sieve, one bit is used to store prime or not prime foreach integer. If one knows that the only even prime is 2 and stores thatinformation in the algorithm instead of in memory, then 1/2 as much memoryis needed. Using the notation above,(2-1)/2 == 1 bit / 2 integersBy not storing 3, then ((2-1)*(3-1))/(2*3) == 2 bits / 6 integers of the storage is required,(or, 1/3 as much storage as was required when one used a bit for each integer) but that is not as convenient for storage at 8 bits/byte.My 8 bits for thirty integers storage scheme could be extended to:(2-1)*(3-1)*(5-1)*(7-1)----------------------- == 48 bits / 210 integers 2*3*5*7which would be 6/7 of storage requirement of the method that I used. Ifextended to the next prime, then only 10/11 as much storage would be used,or 60/77 as much storage as I used.Enter two positive integers in the form below to get a list of primenumbers in the interval. I suggest not doing too large a range to avoidrunning your web browser out of memory. If you want all 98 millionprimes less than 2000000000, then I suggest downloading my sieveprogram. There are links to source code and binaries below. The sieveprogram will run much faster than this slow web server can deliver thenumbers to your browser. For example, my old 300 MHz AMD K6 took 1 hourand 33 minutes just to print the 367783654 primes less than 8000000000.If you want just a few hundred primes, this web server will be fairlyquick and it has all of the primes less than 8000000000. I have done aWindows version of the sieve program that will go to 8000000000, ormaybe further if you have enough ram. The 64 bit program is called sieve2310_64bit.exe. If youwant this for Linux, all that is needed is to change the unsigned longsinto long longs and change the printf formats and compile with gcc3.2 (or newer) or any C99 compliant compiler. sieve2310_64bit.cPlease limit yourself to a range of about 100000 to avoid connection timeouts. If you want more, download the program as mentioned above.The starting integer The ending integer More information about prime numbers from the University of Tennessee at Martin.More information about prime numbers from the University of Utah.An explanation of what prime numbers are from dr.math.Prime numbers as a sequence of integers at www.research.att.com.http://dmoz.org/Science/Math/Number_Theory/Prime_Numbers/ has many good links to web pages about prime numbers.Source code for asieve program from a web server,or embedded in html for easy viewing with Internet Explorersieve2310.c.html or ftp sieve2310.cor to ftp a 32bit ".EXE" file and source for Microsoft Windows sieve2310.exeThis program which uses (with permission) a small amount of my codecalculates various prime number statistics.http://ndirty.cute.fi/~karttu/matikka/Schemuli/primestats.c.txt.Find thenext prime number larger than an integer you provide. Thisis a JAVA program and requires a new enough JAVA in your web browserto include the java.math.BigInteger package. Source code is here.Find thenext ten prime numbers after the integer you provide.factor.exe isa Windows NT commmand line program to print the prime factors of aninteger. It may also work with Windows 98. Type "factor help" toget a usage message. It seems to work on integers up to about9007199254740989. bfactor.exe isanother Windows NT command line program to print the prime factors of aninteger. It will attempt to factor integers up to about 600 digits, but most will take a very long time, maybe many years. If the Windows 98command line is not long enough, bfactor will read one integer per linefrom stdin. 31610054640417607788145206291543662493274686990 is an easyinteger for bfactor.exe to factor if you want to test it without waitingtoo long. There is also a Linuxbinary of bfactor.John Moyer's home pageJohn Moyer's old home page with chemical thermodynamicssend email to John Moyer

Some questions that are frequently asked by visitors to this page.

Where can I find a list of all of the prime numbers?If you were wanting all of the prime numbers instead of just those upto some arbitrary limit, then you are out of luck. There are an infinitenumber of primes. A greek named Euclid proved this a couple thousandyears ago. He proved this by assuming that there were a finite numberof primes and prime N was the biggest prime. Then, one could multiplyall of the primes together and add one to the product. This new numberis not divisible by any of the primes that were multiplied together,therefore it is either a prime itself, or it is a product of a primelarger than the one we earlier assumed to be the largest prime. Sothe assumption was false and there is no largest prime. You cannotget all of the prime numbers. There is always a bigger prime.How can I arrange the digits 1,2,3,4,5,6,7,8,9 as a nine digit prime?If you use each of those digits exactly once, then you cannot arrangethem as a prime. The sum of those nine digits is 45 which is divisibleby 3. Therefore, 3 is a factor of any of those nine digit numbers andnone of them are prime.The reason why this works is that the 9 digit number may be written as apolynomial. To illustrate this with a 3 digit number, 456 may be writtenas 4*10*10+5*10+6. Then, since 10=(9+1) it mayalso be written as 4*(9+1)*(9+1)+5*(9+1)+6. For anarbitrary 3 digit number where the digits are represented by a, b, c, itwould be a*(9+1)*(9+1)+b*(9+1)+c. This may be multipliedout to be (a*9+a*1)*(9+1)+b*(9+1)+c. Multiplying one more time we get ((a*9)*9+(a*1)*9+(a*9)*1+(a*1)*1)+b*9+b*1+c.This simplifies to a*81+a*9+a*9+a+b*9+b+c.Since 3 evenly divides each of the terms that contain a 9 or amultiple of 9, then if 3 evenly divides the remaining terms(a+b+c) the entire number is evenly divisible by 3.Copyright © 1999-2007, John Moyer, All rights reserved.
 

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for

small

numbers.

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primes

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8

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Some Prime Numbers 2008 November

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Sieves and factoring for small numbers. Download primes below 8 billion.

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