# Open conjecture in number theory without a good heuristic?

For most open conjectures in number theory , there are good heuristics that they are true : Examples include the Goldbach conjecture, the Collatz conjecture , the Riemann hypothesis and the twin prime conjecture.

Are there conjectures in number theory which are not disproved, but there is also no good heuristic that they are true ?

• Maybe you mean "experimental evidence" ? – Yves Daoust Apr 3 at 18:35
• May be I think even the results on odd perfect numbers also somewhat belong here. Since it is unknown whether there exists such a number, but there are several results as to the bounds on such numbers – vidyarthi Apr 5 at 7:46
• @vidyarthi In fact, although many mathematicians are convinced that no odd perfect number exists, apart from several restrictions no convincing heuristic is currently known. – Peter Apr 5 at 9:07

The Hardy-Littlewood Conjectures do not have a "good heuristics" and are known to be contradictory to each other. The first one is known as strong twin prime conjecture, and the second one states that $$\pi(x+y)\le \pi(x)+\pi(y)$$ for all $$x,y\ge 2$$.
Fermat primes are prime numbers of the form $2^{2^{n}} + 1.$
They are just too large to check their primality even for moderately small values of $$n$$.