with a , b any coprime integers, a > 1 and − a < b < a . (Since a n − b n is always divisible by a − b , the division is necessary for there to be any chance of finding prime numbers. In fact, this number is the same as the Lucas number U n ( a + b , ab ) , since a and b are the roots of the quadratic equation x 2 − ( a + b ) x + ab = 0 , and this number equals 1 when n = 1 ) We can ask which n makes this number prime. It can be shown that such n must be primes themselves or equal to 4, and n can be 4 if and only if a + b = 1 and a 2 + b 2 is prime. (Since a 4 − b 4 / a − b = ( a + b )( a 2 + b 2 ) . Thus, in this case the pair ( a , b ) must be ( x + 1, − x ) and x 2 + ( x + 1) 2 must be prime. That is, x must be in A027861 .) It is a conjecture that for any pair ( a , b ) such that for every natural number r > 1 , a and b are not both perfect r th powers, and −4 ab is not a perfect fourth power . there are infinitely many values of n such that a n − b n / a − b is prime. (When a and b are both perfect r th powers for an r > 1 or when −4 ab is a perfect fourth power, it can be shown that there are at most two n values with this property, since if so, then a n − b n / a − b can be factored algebraically) However, this has not been proved for any single value of ( a , b ) .
The record passed one million digits in 1999, earning a $50,000 prize.  In 2008 the record passed ten million digits, earning a $100,000 prize and a Cooperative Computing Award from the Electronic Frontier Foundation .  Time called it the 29th top invention of 2008.  Additional prizes are being offered for the first prime number found with at least one hundred million digits and the first with at least one billion digits.  Both the $50,000 and the $100,000 prizes were won by participation in GIMPS.