Quote:
Originally Posted by axn
x+k*f(x) == x (mod f(x))
==> f(x+k*f(x)) == f(x) (mod f(x)). QED
Looks like a trivial result.

Agreed the result is not earthshaking. However it has a couple of interesting applications, one of which I would like to mention in this post:
Indirect primality testing. Let f(x) be a quadratic polynomial in x ( x belongs to Z).
For example let f(x) be x^2 + x +1. All x, other than those generated by 1 + 3k, 2 + 7k, 3 + 13k, 4 + 3k and 4 + 7k, 5 + 31k.....are such that f(x) is prime which need not be tested for primality. Unfortunately this is true only upto quadratic level.