Gilbreath 猜想的进展Progress on Gilbreath’s conjecture
Gilbreath 猜想是一个关于素数的简单却奇特的数学猜想，普通人也能理解其基本逻辑。著名数学家保罗·埃尔德什曾推测该猜想是正确的，但至今仍未得到严格证明。文章回顾了该猜想的历史背景，并探讨了近期在相关领域取得的研究进展。这展示了即使是看似简单的数论问题，也可能蕴含着极深的数学难度。
John
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.
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