Rho:x^2-2

Most polynomials work nicely with Pollard Rho.
However, f(x)=x^2 and f(x)=x^2-2 should be avoided.
Here's what John Pollard had to say by way of reason, in 1975 [BIT 15, P.333]:
"(i) that all polynomials x^2+b seem equally good in (1) except that x^2 and x^2-2 should not be used (whatever the starting value x0), the latter for reasons connected with its appearance in the Lucas-Lehmer test for primality of the Mersenne Numbers [3],"
while Knuth has this to say [TAOCP Vol.2 P.386]:
"In those rare cases where failure occurs for large N, we could try using f(x)=x^2+c for some c<>0 or 1. The value c=-2 should also be avoided, since the recurrence x_(m+1) = (x_m)^2-2 has solutions of the form x_m = r^(2^m) + r^-(2^m). Other values of c do not seem to lead to simple relationships mod p, and they should all be satisfactory when used with suitable starting values."