Exponential speedup of prime sieving
It is surprisingly easy to show that when n equals a primorial (product of successive primes) the inner spacing between co-prime members of Z_n forms a palindrome whose length is one less than the size of the set. Since you can construct the set for the next primorial boundary from your current boundary by expanding the residuals you can double the speed of each new set's construction by taking advantage of this internal symmetry. Meaning at each step you only need to identify half of each new sets co-prime members, the rest is solved by taking advantage of the palindrome.
Please read the paper in the link, I have a lot more equally cool and surprising proofs.