If a number theorist solves a problem that Gauss wasn't able to, the result will get into Annals of Mathematics.
Are there really such problems? Gauss was no Hilbert. If he couldn’t answer a question himself he just kept the failure to himself. The closest thing I know is:
https:// en. wikipedia. org/wiki/Class_number_problem
but I suspect Gauss didn’t explicitly pose it as a conjecture.
My understanding is that there wasn't as much of a culture of posing conjectures and problems in pre-Hilbert days anyway. Most things we would reasonably call conjectures of Fermat or Euler, for example, were just things they wrote down in letters but either didn't prove or explicitly said they weren't able to prove.
In that sense Gauss was more reticent than contemporaries, but the prime number theorem is a good example of a statement he said he believed but wasn't able to prove.