What all are still Unproven Conjectures in Number Theory

Question

I know that Goldbach's conjecture is yet to be proved even though it has been shown to hold up through $4 \times10^{18}$. I haven't been able to find a comprehensive list of unproven conjectures on google.There is a list of unsolved problems, list of conjectures but not that of unproven ones.I am interested in knowing if some conjectures(related to Number Theory), that are yet unproven, are still extensively used while arriving at solutions of mathematical problems.

Answer 1

One I recently discovered is the conjecture that all perfect numbers are even. The first $10^{300}$ natural numbers have been checked, and all perfect numbers in that range are even.

Answer 2

This Wikipedia page lists most of the important ones. It's how I first got interested in research mathematics. All of the ones listed in the main body of the page are unproven so far.

In terms of the ones that are used a lot by mathematicians, I would have to say that the Riemann Hypothesis is on top, as it is quoted that "Entire books are written assuming the Riemann Hypothesis" (I think that's Derbyshire).