Does anyone here know a good way to come up with good mathematical conjectures that are likely true? I don't have too much experience with this myself, but I know that some mathematicians are experts at this. Paul Erdos, for one, was able to come up with over a thousand number theory conjectures and prove about half of them. Although I haven't come up with too many myself, much less proven any of them, I'd say a big criterion is that if some mathematical fact is true, especially if it seems surprising or counterintuitive, then there's usually a good reason for it. For instance, why should there be a larger fraction of primes congruent to 1 mod 4 than to 3 mod 4? Although this is quite difficult to prove, it seems pretty obvious to me, because what's so special about either modulus? Another example is the twin primes conjecture, since prime gaps seem pretty random, other than the fact that they're all even except for the first one, so why should there be only finitely many equal to 2?…