# Can the openness of Gilbreath's conjecture be reduced to proof of the Riemann hypothesis?

As an information theoretician, it's a personal hobby of mine to find elegant analogies to open mathematical problems. After all, they have a profound impact on how I conduct my research and how I secure my systems from theoretical attacks.

While playing with Gilbreath's conjecture, I noticed, as many do, that the conjecture might be solved trivially if one could reason exactly about an infinite sequence of prime numbers. Because all other aspects of the problem are known (including the properties of the forward difference function) and because proof of the Riemann hypothesis can be reasoned to be hypothetically as difficult as a solution to the conjecture (through the nature of prime numbers), can a variation of Gilbreath's conjecture be shown to be solved inductively by application of the Riemann zeta function? If so, what implications does this have?

I'm probably well off base here, and this property is probably well known. But, my curiosity is insatiable. Links to supporting information or other questions may well be sufficient to answer this conjecture.

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The logical extension of your question is, if you have any two unsolved problems, can you use one to solve the other? You haven't given any reason to think there's any connection between Gilbreath and Riemann, other than that they're both difficult and have something to do with primes - but there are a thousand difficult questions about primes, and no one has the time to decide for each pair of such questions whether one has some effect on the other. Anyway, you've linked to the wikipedia pages - does either one mention the other? If not, what does that suggest? – Gerry Myerson Nov 5 '11 at 3:31
In this case, I'm trying to hinge both problems on reasoning about the nature of the gap between primes. Proof of the Riemann hypothesis would enable the conjecture to be strongly estimated, by significantly reducing the relative error in an arbitrary sequence of estimated primes, allowing an inductive proof. Conversely, exact proof of the nature of the primes would, to my knowledge, solve both the hypothesis and Gilbreath's conjecture. As for your comment about Wikipedia, the link is [Gilbreath's_conjecture]->[Prime_gap]->[Prime_number_theorem]->[Riemann_hypothes‌​is]. – MrGomez Nov 5 '11 at 6:32
@GerryMyerson Although, I agree with you insofar as this is very likely an intellectual fishing expedition. The concreteness of my question stems from whether Gilbreath's conjecture can be shown to work through Riemann's zeta function, which would tell us the two problems may well be linked, improving the information for both and satisfying my curiosity for the former. I don't think there's a chance in $YOUR_FAVORITE_ANALOGUE_FOR_HELL_HERE that we'll be solving either problem in the context of this thread. – MrGomez Nov 5 '11 at 6:40 If we knew that there's always a prime between$n$and$n+2(\log n)^2\$ (which is far stronger than anything implied by the Riemann Hypothesis), would that imply Gilbreath? If we knew Gilbreath, would it give us a bound on prime gaps strong enough to imply Riemann? I don't know what it means for a conjecture "to be strongly estimated." I don't know what "exact proof of the nature of the primes" means. You have to use well-defined, meaningful terms if you are going to communicate something (much less prove something). – Gerry Myerson Nov 5 '11 at 9:01
@GerryMyerson And, while I agree, the answer to your question of semantics is a firm "I do not know how to better frame this question, because I am not a mathematician." My question, then, isn't one of research -- it's one of pre-existing sources not covered by the common content portals that I might use to further my layperson's understanding. – MrGomez Nov 5 '11 at 21:45