the fibonacci numbers are the set of positive values of this polynomial, where x and y are > 0: g(x, y) = 2 x y^4 + x^2 y^3 - 2 x^3 y^2 - y^5 - x^4 y + 2 y (ribenboim, the new book of prime number records, p. 193) it is easy to find positive (x,y) that give positive values of g. for example, if x = F(n-1) and y = F(n), then g(x, y) = F(n). similarly, the prime numbers are the set of positive values of a 25th degree polynomial in 26 variables. the polynomial is given here: http://primes.utm.edu/glossary/page.php?sort=matijasevicpoly see also http://mathworld.wolfram.com/PrimeDiophantineEquations.html this means that, for any prime P, there exist positive integers (a, b, ..., z) that, when plugged in to the big polynomial, give P. it also means that, if you can find (a,b,..z) that give a positive value Q of the polynomial, then Q is prime. the big polynomial reduces to a bunch of simultaneous diophantine equations, each with a few variables, and of degree about 2 - 6. has anyone seriously tried to solve these equations to find one such explicit representation of a prime (either for a fixed P, or just finding any (a,b,...z) that give a positive value of the polynomial? bob