GIMPS Sets Another Record! 22976221-1 is prime

By Chris Caldwell

There is no getting around it, the 895,932 digit number 22976221-1 is a giant--it would easily fill a 400 pagepaper back book!  Nevertheless, on August 24th, 1997 Gordon Spence completed the proof of its primality on a 100 MHz Pentium.  The complete proof took fifteen days using a highly optimized program written by the founder of the Great Internet Mersenne Prime Search, George Woltman. David Slowinskiof Cray Research finished verifying the primality on August 29th. It is now the largest known prime, the 36thknown Mersenne Prime, gives rise to the 36th known perfect number (with 1,791,864 digits!) and smashes all previous records for size

How do you show a number that large number is prime?

A prime is an integer that has only one and itself for positive divisors (for example, the prime factors of 10 are 2 and 5).  The first few primesare 2, 3, 5, 7 and 11.  We can check small numbers by just dividing by the lesser integers.  But to do that with a number this size would take far longer than the universe has been in existence.  The key to slaying a giant this size is a theorem developed by Lucasin the late 1870's and simplified by Lehmer, now called the Lucas-Lehmer Test.  Even as simple as this test is, it could not be done quickly without using very clever programs to multiply the numbers. In 1968 Strassen discovered how to multiply quickly using Fast Fourier Transforms (*). He and Schönhage refined and published the method in 1971. GIMPS now uses an improved version of their algorithm developed by the long time Mersenne searcher Richard Crandall[see CF94]. 

Is that it, just get a fast program?

Even a fast program is not enough, alone a systematic search would take nearly a millennium on Spence's computer.  What GIMP's does is to coordinate the efforts of over two thousand computer users--by offering free software, a database of known results, and by allowing individuals to check out regions to test.   Gordon Spence and  George Woltman share the credit and glory with all of the others involved in this effort!  There are still infinitely manymore giants left to slay, so why not surf over to Woltman's GIMPS siteand join the search for the next record prime?

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