with , any coprime integers, and . (Since is always divisible by , 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 , since and are the roots of the quadratic equation , and this number equals 1 when ) We can ask which makes this number prime. It can be shown that such must be primes themselves or equal to 4, and can be 4 if and only if and is prime. (Since . Thus, in this case the pair must be and must be prime. That is, must be in .) It is a conjecture that for any pair such that for every natural number , and are not both perfect th powers, and is not a perfect fourth power. there are infinitely many values of such that is prime. (When and are both perfect th powers for an or when is a perfect fourth power, it can be shown that there are at most two values with this property, since if so, then can be factored algebraically) However, this has not been proved for any single value of .