Years ago I wrote about Gilbreath’s conjecture. It’s a simple conjecture; you could explain it to anyone who understands what prime numbers are. See the linked post for a description of the problem.
Gilbreath’s conjecture is simple, but it’s also kinda weird. As I wrote before,
Paul Erdős speculated that Gilbreath’s conjecture is true but it would be 200 years before anyone could prove it. I find Erdős’s conjecture more interesting than Gilbreath’s conjecture.
The conjecture is hard in a way that, say, solving a nasty-looking differential equation is not. Over the last three centuries, mathematics has developed quite a toolbox for solving differential equations. But Gilbreath’s conjecture is just odd enough that it’s not at all what kind of tool might be useful in approaching it.
Terence Tao has a new blog post announcing a paper that he and two coauthors wrote on a random model intended to mimic Gilbreath’s calculation on primes. This random model is more sophisticated than the little game Gilbreath was playing, but it’s also much more amenable to analysis by established techniques. Tao’s post gives a heuristic explanation for why Gilbreath’s conjecture is plausible, but then adds
However, it seems well beyond current technology to try to make these heuristics rigorous; even the first step … is far out of reach.