GPT-6 Astra and Amateur Mathematician Prove 57-Year-Old Circle Packing Problem

The problem being advanced here is not itself complicated: given N disks of radius 1 and a large circle of radius R, all small disks must lie entirely within the large circle and must not overlap in their interiors with one another (tangency is allowed); find the minimum value of the large circle's radius R.

The difficulty lies not in finding an arrangement, but in proving that a certain arrangement has reached the limit. Mathematicians gave an elegant packing arrangement as early as 1969, but for 57 years no one has been able to rigorously prove its optimality.

Historical progress has been slow: in 1968 the American mathematician Graham proved N≤7; in 1969 the mathematics researcher Udo Pirl advanced the proof to 10 disks and further studied packing arrangements for 11 to 19 disks; it was not until around 2003 that the cases N≤14 and N=19 were rigorously proven. In other words, before this work, only N≤14 and N=19 had been rigorously verified.

According to a QbitAI report, the math enthusiast nor used GPT-6 Astra to rigorously prove, in one go, the five cases N=15, 16, 17, 18, and 20 — that is, the remaining cases for N≤20. If valid, all cases for N≤20 would then be proven.

The report states that nor proceeded step by step: first having GPT-6 Astra handle N=15, and after the model solved it, having it extend the same method to 16, 17, as well as 18 and 20, with the model providing proof results for all.

It must be made clear that the above phrasing of "rigorous proof" comes from nor's own blog and QbitAI's relay report; it is a source claim, not an independently confirmed fact. No independent third-party verification or peer review information regarding this proof appears in the given sources.

The report summarizes GPT-6 Astra's core approach as follows: rather than directly searching for the optimal solution, it uses geometric constraints to reduce infinitely many possible arrangements to finitely many cases, then uses a computer to eliminate them one by one, ultimately rigorously proving that the classic arrangement is already optimal.

The report further condenses the proof process into 5 steps: push the outer disks toward the boundary; use angular relationships to eliminate a large number of arrangements; have the computer enumerate the remaining possibilities; use geometric rigidity to prove exact optimality; and restore the original arrangement. The 15-disk case used a different odd-cycle argument.

The report also acknowledges that the proof process is obscure and puzzling, and that the above 5 steps are only a summary; it does not provide verifiable complete proof details. Therefore, whether these 5 steps fully reflect the actual proof structure remains uncertain.

The report emphasizes that this was not a case of nor giving one round of prompts and Astra completing the entire proof. nor wrote in his blog that his contributions as a human included: determining the problem's feasibility, guiding the model, struggling with a bug-ridden web user interface, enduring terrible copywriting, fixing minor issues, repeatedly refining the exposition, manually verifying validity, and using Astra and Opus 5.5 to recheck everything.

This self-account itself also indicates that the result depends on manual review. The report does not state the specific version of GPT-6 Astra, the runtime configuration, or the conditions under which the proof can be reproduced.

As for the claim that "N reaching 20 is not the AI's limit, it is just that nor did not think to continue proving further," this comes from nor's personal choice; the report provides no boundary tests for continuing the proof, so no inference about the model's capability ceiling can be drawn from it.

The given source consists of only one QbitAI report, whose content mainly relays nor's own blog and X posts, lacking a first-party original paper or independent reproduction. The original links given in the report are nor's blog post (nor-blog.pages.dev) and nor's post on the X platform.

The resulting uncertainties include: whether the proof has undergone independent verification or peer review by the mathematics community; whether the proof given by GPT-6 Astra is fully rigorous and whether there are undiscovered gaps; and whether the specific content of nor's blog and X posts as first-party sources is fully consistent with QbitAI's relay.

In addition, the specific differences between the odd-cycle argument used for N=15 and the methods for the other cases are only mentioned in passing in the source; the source also provides no evidence as to whether this result will be reproduced by subsequent research or generalized to N>20.

The report mentions an interesting sidelight: after nor published the result, he found that there were quite a few like-minded people using AI to study the same problem, and some netizens said that AI breakthroughs have made mathematics easier to understand.

This is a community observation at the level of a source claim, and its scale or representativeness cannot be further verified from the given materials. But it at least suggests that such attempts at AI-assisted mathematical proof are appearing in the form of personal blogs and social platform posts, rather than through traditional paper channels.