The reason why this works is that the 9 digit number may be written as a polynomial. To illustrate this with a 3 digit number, 456 may be written as 4*10*10+5*10+6 . Then, since 10=(9+1) it may also be written as 4*(9+1)*(9+1)+5*(9+1)+6 . For an arbitrary 3 digit number where the digits are represented by a, b, c , it would be a*(9+1)*(9+1)+b*(9+1)+c . This may be multiplied out 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 a multiple of 9, then if 3 evenly divides the remaining terms (a+b+c) the entire number is evenly divisible by 3.
As promised, GIMPS gave $50,000 of the EFF award to the UCLA Department of Mathematics, where Edson Smith was responsible for installing and maintaining the GIMPS software on their computers. Another $25,000 has been donated to a math-related charity selected by GIMPS founder George Woltman. The remaining $25,000 has been paid in GIMPS Mersenne Prime Research Discovery Awards to Odd Magnar Strindmo for his discovery of M47 , Hans-Michael Elvenich for M46 , the University of Central Missouri ( M44 and M43 ), Dr. Martin Nowak ( M42 ), Josh Findley ( M41 ), Michael Shafer and his selected charity ( M40 ) and Michael Cameron ( M39 ).
A particularly simple example of a probabilistic test is the Fermat primality test , which relies on the fact ( Fermat's little theorem ) that n p ≡n (mod p) for any n if p is a prime number. If we have a number b that we want to test for primality, then we work out n b (mod b) for a random value of n as our test. A flaw with this test is that there are some composite numbers (the Carmichael numbers ) that satisfy the Fermat identity even though they are not prime, so the test has no way of distinguishing between prime numbers and Carmichael numbers. Carmichael numbers are substantially rarer than prime numbers, though, so this test can be useful for practical purposes. More powerful extensions of the Fermat primality test, such as the Baillie-PSW and Miller-Rabin tests, may fail at least some of the time when applied to a composite number.