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 ?

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    $\begingroup$ Maybe you mean "experimental evidence" ? $\endgroup$ – Yves Daoust Apr 3 at 18:35
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    $\begingroup$ 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 $\endgroup$ – vidyarthi Apr 5 at 7:46
  • $\begingroup$ @vidyarthi In fact, although many mathematicians are convinced that no odd perfect number exists, apart from several restrictions no convincing heuristic is currently known. $\endgroup$ – 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 $x,y\ge 2$.


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 $n$.


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