M34

M34
Prime class :
Type :	Mersenne prime
Formula :	Mn = 2n - 1
Prime data :
Rank :	34
n-value :	1,257,787
Number :	412245773621...976089366527
Digits :	378,632
Perfect number :	21,257,786 • (21,257,787-1)
Digits :	757,263
Discovery data :
Date of Discovery :	1996-09-03
Discoverer :	David Slowinski Paul Gage
Found with :	Lucas-Lehmer test / Cray T94

The 34th Mersenne prime, both in order of size (smallest to largest) and in order of discovery. Specifically M34 is 2^1,257,787-1, which is a number 378,632 decimal digits long. The number was found to be prime in 1996.

David Slowinski and Paul Gage running Lucas-Lehmer test software that they had written (known as mprime) on a Cray Research T94 supercomputer in a run of ~6 hours discovered that the number is prime.

It was verified by Richard Crandall as well as by George Woltman and others. Using the Prime95 software on a Pentium 90MHz machine it took about 60 hours to double check the number. Woltman, by happenstance, was 95% of the way through a check on that very number, when he was notified of Slowinski and Gage's result (on April 15) and asked to verify it. This being the case, Woltman's verification was the first to be completed (in about the same length of time from his notification to the end of the test as Slowinski & Gage's test took.) Had Woltman's test been completed before the notification from Slowinksi arrived, he would have been a co-discoverer. Had Woltman been testing the number on his Pentium-Pro 200MHz machine, he may have been the first to notify the others of the discovery.

This was the last record Mersenne prime to be discovered on a supercomputer and not on a personal computer. The above incident shows how close it came to being the first record Mersenne prime to be discovered on a PC. This parallels the general industry trend from very high performance specialized processor based systems to highly parallel 'off-the-shelf' chip base systems. Many of the fastest supercomputers now are GPUs and server processors.

Cray and Silcon Graphics (SGI) which purchased Cray had employees who, either while with the company or before they joined the company, discovered 10 of the 11 record primes between 1978 and 1996, namely Slowinski, Gage, and Landon Curt Noll (either alone or with others).

Why would such a company allow their valuable computer time to be 'wasted' on this search?

In a press release about the discovery of M34 the company said, "the 'prime finder' program developed by Slowinski and Gage is used by Cray Research as a quality assurance test on all new supercomputer systems." Slowinski said, "This acts as a real 'torture test' for a computer. The prime finder program rigorously tests all elements of a system -- from the logic of the processors, to the memory, the compiler and the operating and multitasking systems. For high performance systems with multiple processors, this is an excellent test of the system's ability to keep track of where all the data is."

So, in addition to checking the machines, it also helped with mathematics research.

Sources

• The Prime Pages
• Noll's website