Prize for next Goldbug or proof no more exist

[Craig Beisel]

A few people have attempted but so far no one has been able to find the next Goldbug Number. 😥 I really want to know if it's out there so I'm going to try and double the prize every year. This means I am now offering $4000 to the person(s) who identify the next number OR prove there isn't one. Check out the OEIS entry A306746 for more information.

[Ruud H.G. van Tol]

As a side remark, the entry mentions: a(7) > 5*10^8,
but likely a(6) is meant there.
(unless 2200 would count again)

-- Ruud

[Craig Beisel]

Yes good catch. I think the confusion is due to the fact that 2200 is counted as a basic pipe by Wu. I will put in a fix to that entry, also it looks like Wu's page is gone at this point which is a shame. I will contact him and see if I can get his paper.

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[jpallouche.math]

Hi, my two cents:

https://web.archive.org/web/20201017182439/https://sites.google.com/site/basicpipetheory/

https://web.archive.org/web/20201017182440/https://sites.google.com/site/basicpipetheory/doc

but, unfortunately, on the last one, most of links were not saved, with the exceptions of old versions:

https://web.archive.org/web/20201017182450/https://170f119b-a-62cb3a1a-s-sites.googlegroups.com/site/basicpipetheory/doc/pipe2010_01.pdf?attachauth=ANoY7cqeHJux28AvwLBDDjf8jcgFu9HoJ0OlPH2fGdDenpTiBxpLCSJhHOHzk3nO8YIJBBCqd_nxzzVnyDjA10YN41PlXPJE7izUrbMt3FkgihNdfUXuFjP_K19XkfKzMsYVgmc6mRyfKl4GJr3JfbO-cYbx4YCtF4imj4ZAnr_yUXHSLjbVloFUekl5CsUQdBygL5Q-C8Mlx2b8fRZ4kdurfglmcr6OjW-1VRiY7dBlZZs8Fpn9ur0%3D&attredirects=0&d=1

and

https://web.archive.org/web/20201017182446/https://170f119b-a-62cb3a1a-s-sites.googlegroups.com/site/basicpipetheory/doc/pipe2009_10.pdf?attachauth=ANoY7cofvc30uvhz_yGFP666wYVLKr6AM8xymwPjHo_EL4WU8_4ELo3Te3ukxf8bsFv8fx15KVuR0TFdleLbuel8g1LK_GmriXw2be7KWN2TGrbjfj-dKmADe2bLLHK2OhmTc0sqfSqGxl8kVlY3GUbJHWI3Y0gGSOsJIpg94gte_KiG5WD6Ki5jA9Ugbe8_Xom5k0CfdnnUCn8zGFyYQGXkBS7iKe2gN5XTP73jy7ct2Ddxves9YlQ%3D&attredirects=0&d=1

best wishes
jean-paul

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[Arthur O'Dwyer]

Word to the wise: Exhaustive search is probably not the way to go, at this point.

I put the Python code from that Math.SE post here, with my own C++ translation; the latter is not appreciably faster.

Wu's "Introduction to Pipe Theory" (thanks, Jean-Paul) lists some candidate Goldbugs such as the 173-digit number

14210043931054384697281797631385857688949646035466487917230047060794326605647294255025896867407138484058379599443343129460393572580187625386158635949233364384758021881468840

but I myself wouldn't know how to go about proving or disproving that number's Goldbug-ness or even its Goldbach-ness. :)  (I'm sure people with more math and more fluency in mathematical programming will find at least one of those tasks easy; I bet that that number is the sum of two primes (easy?) and is not a Goldbug (hard?).)

Craig, does your bounty apply to the finder of any new Goldbug, or only to the provably next Goldbug? Presumably the former, but I suppose that detail might affect some readers' strategies. ;)

(Not me; I've satisfied myself that I've got no chance here.)

–Arthur

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[Craig Beisel]

The $4k bounty is for identifying the next one which is going to be large if it exists. Easy to verify but hard to find. I had forgotten that Wu had named some candidates, that might be helpful if he was right ;^) I have looked pretty high but I believe Wu searched higher  to 5x10^8. Obviously we run into a combinatorial explosion as we increase the number of primes. But maybe there is a clever way I havent thought of. Here is a link to the best code I have seen which I did not author myself...

https://math.stackexchange.com/questions/3026044/what-is-the-best-algorithm-for-finding-goldbug-numbers

Also I will pay for proving there are no additional goldbugs, but doesnt this effectively prove goldbach since it identifies an algorithm that always finds a prime pair above a certain threshold? I'm going to guess that this is even harder than finding the next number, but who knows since it's a slightly different problem than proving goldbach directly.

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[Craig Beisel]

It goes without saying but I guess technically I should say that I pay out only if the next number in the sequence is published on OEIS and goes through review. 

[Craig Beisel]

Actually I see your point. What I am interested in knowing is if there are additional goldbigs. So you are right, I will change the prize to include finding any goldbugs greater than 6142. It does not need to be the next one.

Hide quoted text

[Craig Beisel]

I finally see Arthur's point and I agree. Prize pays out for any discovered Goldbug, it does not need to be the next one necessarily.

[Max Alekseyev]

Here is the latest (Dec 2019) version of Wu's work:

https://web.archive.org/web/20201017182601/https://170f119b-a-62cb3a1a-s-sites.googlegroups.com/site/basicpipetheory/doc/Willie%20Wu%20Goldbach%20Conjecture%20Pipe%20Theory%20Dec%202019.pdf?attachauth=ANoY7cqQ7NWmi40OLI01yLEnSJJnzbjzFRsbMw_VBRReu4eeQ4HuIqAOmC7-joYUXXgSI2lEORwGXTXstGPtdZWLe3nmtVYvF7ApYvth97d5cdtXtCpWW_LIAUrXPuggQcMGTxJl2sTNgvljyt-IA6lftD-OYOgJ8hhvhBy-5jswKm7XbmvGS3eIHNgsBp2qA6HLIITtI_8xt9qE-qQhwxOxPlGe9EzsOrSTQ9fsvi4hLVNY_N4iKM4oYNZGHt8MxFyVFzU2BuRr_67vMgVMoTJeX-Mr0rPKbg%3D%3D&attredirects=0

Regards,

Max

On Mon, Feb 23, 2026 at 12:50 PM jpallouche.math <jpallou...@gmail.com> wrote:

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