There are a huge number of examples like that floating around in the literature, we link to some of them in the writeup. I think Terence Tao’s blog is the easiest place to get an overview of these arguments, see this post in particular but he discusses this kind of reasoning often.
I think it’s easy to give probabilistic heuristic arguments for about 80 of the ~100 conjectures in the wikipedia category unsolved problems in number theory.
About 30 of those (including the Goldbach conjecture) follow from the Cramer random model of the primes. Another 9 are slightly non-trivial applications of random models for the primes. About 8 of them are simple heuristics for diophantine equations (like Fermat’s last theorem). I estimate that another ~30 have arguments that are more diverse and interesting (I estimated the size of this set by randomly sampling some conjectures, stratified by difficulty, and seeing how often we could find an argument in an hour).
We’d guess that it’s also possible to give arguments for the remaining ~20, it would just be too hard for us to do within an hour. Random representative examples for which we don’t know heuristic arguments, sorted by apparent difficulty for a layperson:
The surprisingly low density of solutions to Lehmer’s totient problem (link)
The existence of an incongruent covering system with odd moduli (link)
We are interested in the conjecture that all true statements are have a probabilistic heuristic argument for plausibility, and this is a possible source of counterexamples to that conjecture.
This is a nice suite of test cases to see if a proposed formalization of probabilistic heuristic arguments captures the kinds of arguments that are intuitively valid.
That said, I think that many people believe that number theory is an unusual domain that is particularly amenable to probabilistic heuristic arguments, and so it’s likely not the best place to search for counterexamples.
Great work!
Stuart Armstrong gave one more example of a heuristic argument based in the presumption of independence here.
https://www.lesswrong.com/posts/iNFZG4d9W848zsgch/the-goldbach-conjecture-is-probably-correct-so-was-fermat-s
There are a huge number of examples like that floating around in the literature, we link to some of them in the writeup. I think Terence Tao’s blog is the easiest place to get an overview of these arguments, see this post in particular but he discusses this kind of reasoning often.
I think it’s easy to give probabilistic heuristic arguments for about 80 of the ~100 conjectures in the wikipedia category unsolved problems in number theory.
About 30 of those (including the Goldbach conjecture) follow from the Cramer random model of the primes. Another 9 are slightly non-trivial applications of random models for the primes. About 8 of them are simple heuristics for diophantine equations (like Fermat’s last theorem). I estimate that another ~30 have arguments that are more diverse and interesting (I estimated the size of this set by randomly sampling some conjectures, stratified by difficulty, and seeing how often we could find an argument in an hour).
We’d guess that it’s also possible to give arguments for the remaining ~20, it would just be too hard for us to do within an hour. Random representative examples for which we don’t know heuristic arguments, sorted by apparent difficulty for a layperson:
The surprisingly low density of solutions to Lehmer’s totient problem (link)
The existence of an incongruent covering system with odd moduli (link)
The Birch and Swinnerton-Dyer conjecture (link)
The Grothendieck-Katz p-curvature conjecture (link)
Serre’s Conjecture II (link)
This category is interesting to us because:
We are interested in the conjecture that all true statements are have a probabilistic heuristic argument for plausibility, and this is a possible source of counterexamples to that conjecture.
This is a nice suite of test cases to see if a proposed formalization of probabilistic heuristic arguments captures the kinds of arguments that are intuitively valid.
That said, I think that many people believe that number theory is an unusual domain that is particularly amenable to probabilistic heuristic arguments, and so it’s likely not the best place to search for counterexamples.