Taking all $C(900,2)$ combinations of the first 900 prime numbers, I defined $N = pq$, where $p$ and $q$ are a combination of primes. Then I factored $N$ using Pollard's Rho, counting how many iterations and how many successes were found, where a success is finding a non-trivial factor and an iteration is a computation of the GCD algorithm.
Notice how even powers have much better performance while keeping about the same number of successes. Can you explain that?
iterations sucesses polynomial
784491 44081 x^2 + 1, x0 = 2
831938 44098 x^2 + 1, x0 = 3
3380606 44505 x^3 + 1, x0 = 2
3176223 44511 x^3 + 1, x0 = 3
566286 43757 x^4 + 1, x0 = 2
581990 43765 x^4 + 1, x0 = 3
5912991 44646 x^5 + 1, x0 = 2
6050589 44622 x^5 + 1, x0 = 3
486181 43630 x^6 + 1, x0 = 2
507281 43707 x^6 + 1, x0 = 3
7514619 44717 x^7 + 1, x0 = 2
7884300 44727 x^7 + 1, x0 = 3
528307 43822 x^8 + 1, x0 = 2
528177 43809 x^8 + 1, x0 = 3
3139981 44455 x^9 + 1, x0 = 2
3027514 44430 x^9 + 1, x0 = 3
552764 43801 x^10 + 1, x0 = 2
568909 43823 x^10 + 1, x0 = 3
8623590 44741 x^11 + 1, x0 = 2
8463129 44743 x^11 + 1, x0 = 3
400465 43396 x^12 + 1, x0 = 2
401493 43316 x^12 + 1, x0 = 3
8497977 44733 x^13 + 1, x0 = 2
8047224 44761 x^13 + 1, x0 = 3
588730 43910 x^14 + 1, x0 = 2
634799 43911 x^14 + 1, x0 = 3
1686343 44145 x^15 + 1, x0 = 2
2091292 44238 x^15 + 1, x0 = 3
479840 43619 x^16 + 1, x0 = 2
506561 43579 x^16 + 1, x0 = 3
8919461 44766 x^17 + 1, x0 = 2
8673652 44747 x^17 + 1, x0 = 3
448822 43469 x^18 + 1, x0 = 2
461822 43562 x^18 + 1, x0 = 3
8782917 44766 x^19 + 1, x0 = 2
8466664 44768 x^19 + 1, x0 = 3