Paradoxes in number theory Does it exist any paradoxes within the field of number theory? Any examples?
My thought is that since it is possible to find paradoxes in set theory, which in some sense more fundamental than number theory it should perhaps also exist paradoxes in number theory? However, I have not been able to find any.
Is my reasoning flawed?
 A: A paradox, in the precise meaning of the word, is a statement that is proven to be true and false at the same time. No such statement is knows to exist in mathematics (set theory or elsewhere). There is also no proof that no paradoxes exist, so, as far as we know, paradoxes could exist and we just did not find them, or they do not exist. It is a famous theorem in logic that (under some conditions) it is impossible to prove no paradoxes exist. So, essentially, the situation is that we don't know of any paradox, though paradoxes could exist. If paradox do not exist however, then we will never be able to prove that that is the case. 
Perhaps you meant to use the more colloquial interpretation of paradox, meaning a statement that may appear intuitively to be true (or false) while in fact it is false (or true). Such statements certainly exist in set theory and other areas of mathematics. Of course, what one considers intuitive true another may consider blatantly false, so this is a matter of taste. 
Here are a few such 'paradoxes' from number theory, it's just that I would call them surprising rather than paradoxical. 
1) The fact that there is a bijection between the natural numbers and the squares. This may be considered set theory, but in any case it perplexed many serious thinkers. 
2) The fact that there are lots of integer solutions to $x^2+y^2=z^2$, and all of them are known, while there are no non-trivial solutions to $x^n+y^n=z^n$ for $n\ge 3$, while the proof of that is tremendously difficult (this is Fermat's Last Theorem).
3) The existence of Carmichael primes may come as a surprise. Alternatively, the fact that the smallest Carmichael prime is very small (561) may be surprising. 
4) The Moebius inversion formula if not surprising is at the very least extremely beautiful. 
5) The relation between the prime numbers and the zeta function is quite amazing. 
