Open conjecture in number theory without a good heuristic?

Question

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 ?

Answer 1

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 $x,y\ge 2$.

Answer 2

How about the infinitude of the number of Fermat primes?

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 $n$.