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M36 is the short hand used to refer to the 36th Mersenne prime, specifically it is [math]2^{2976221}-1[/math]. This number was dicovered to be prime on 1997-08-24 by Gordon Spence, using Prime95 written by George Woltman. At time of its discovery, it was the largest known prime number. The number is 895,932 decimal digits long, more than twice the length of the previous record prime! If printed, the number would fill a 450 page paperback book. It took Spence's 100 MHz Pentium computer 15 days to prove the number prime. Alan White Managing Director at Technology Business Solutions, who provided the historic PC, said "We were delighted to donate the computer that has made this exciting discovery."

The primality was independently verified on a Cray T90 supercomputer by David Slowinski, discoverer of seven Mersenne primes between 1979 and 1996.

This prime number was the second record prime found by the GIMPS project, thus demonstrating the power of distributed computing projects. Discovering a prime number of this size would have been impossible without the joint effort. GIMPS is an example what can be accomplished when people, using spare computer time that would otherwise be wasted, combine forces over the Internet. Working alone, it would have taken Spence's computer 940 years to find this prime number.

The corresponding perfect number is [math]2^{2976220} * (2^{2976221}-1)[/math]. This number is 1,791,864 digits long.