A Mathematical Conjecture
Let the index of the ith prime be i, i.e., the index of 2 is 1, the index of 3 is 2, etc.
If we take the prime factorization of an integer n greater than 1, the index of at least one of the primes will be relatively prime to n.
This can be easily proved using ZFC set theory. (Hint: Consider the enumeration of the ordinals less than ε0 discussed here and then recall the Axiom of Foundation.) I'm not sure if it can be proved from the Peano postulates. An enumeration of ε0 cannot be derived from the Peano postulates, but there might be other proofs.
Addendum: There is another proof.