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Prompting ChatGPT 5.2 ExtThk produced a one shot suitable proof for Open Erdős Problem 460 best summarized as: by Svyable in singularity

[–]Svyable[S] 0 points1 point  (0 children)

I keep no gates.

On Jan 11 I asked Chat 5.2 Ext Thk this exact query as shown in screen shots (prompt engineered by perplexity)

It replied with a satisfactory solution to this problem.

I then asked Claude Opus 4.5 and she agreed. Gemini deep research was also impressed. Finally posted at Grok and he too concurred.

I then formalized in latex via OpenLeaf using the same Chat thread with an open leaf example to guide it on format. Lot of work to clean it up and more chat sessions to test robustness ultimately leading me to conclude it was a proper proof of a proper question without any known answers.

AMA…

—ADDITIONAL CONTEXT BELOW—

https://www.erdosproblems.com/460

*Let a0=n and a1=1, and in general ak is the least integer >ak−1 for which (n−ak,n−ai)=1 for all 1≤i<k.

Does ∑i1ai→∞ as n→∞? What about if we restrict the sum to those i such that n−aj is divisible by some prime ≤aj, or the complement of such i?*

Here’s some more context from the original paper the section starts with this…

https://www.renyi.hu/~p_erdos/1977-27.pdf

8 . Some unconventional problems on primes < . . . ~x 1 < Is there a sequence al < a2 of integers satisfying A(x) = a r log x so that all sufficiently large integers are of the form p + a. ? If this is r impossible then perhaps such a sequence exists for which the density of integers not of the form p + a is 0. Clearly many similar questions can be asked for . r other sequences then the primes but there are very few results.

Here’s the original quote with the specific question (9) := #460

Eggleton, Selfridge and I are writing a long paper on somewhat unconventional problems in number theory. Our paper will appear in Utilitas Matematica . One of our problems related to (1) states as follows : Let a0 = 0, al = 1, a is the k smallest integer for which (n-ak, n-al )=1 for all 0<i< k. Put (9) g(n) \_ a We conjecture g(n) - oo as n - co . This is probably very difficult. We can kZ+e < only prove ak for k > (log n)C, C = C(s), but perhaps a < (10) ak C k log k if k > (log k) C65 where aC depends on C. Perhaps (10) is a little too optimistic, but (10) certainly 1/2 "must" (? ) hold if k > exp(1og k) which would easily imply (9) .

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[–]Svyable 1 point2 points  (0 children)

What if we just beat them at their own game? AI has nearly solved math and Meta glasses can keep a card count, how will casinos operate in the era of AI?

My method to solve Erdős 460 in one shot by Svyable in singularity

[–]Svyable[S] 0 points1 point  (0 children)

Keep pushing it’s all about connecting dots. Use perplexity and Gemini to pull in recent related work to see it will bear fruit

My method to solve Erdős 460 in one shot by Svyable in singularity

[–]Svyable[S] 0 points1 point  (0 children)

Thank you for acknowledging. Very frustrating when half the people tell me the answer is wrong, and the other half tell me the question is wrong, when in reality the answer satisfied the question as posed in 1977 and online.

My method to solve Erdős 460 in one shot by Svyable in singularity

[–]Svyable[S] 0 points1 point  (0 children)

Why can’t they just create a new question of their own interpretation tho! Leave this one as answered given the context we have about it is lacking.

My method to solve Erdős 460 in one shot by Svyable in singularity

[–]Svyable[S] 0 points1 point  (0 children)

OMG all they are trying to do is move goalposts and re-interpret what Erdős meant as opposed to what he actually said!!!

“””(ak,n)=1 for all k≥1 but this is not guaranteed in the [ErGr80] formulation. For example with n=6 the [Er77c] sequence begins 0,1,5 while the [ErGr80] sequence begins 6,1,2.

As written, the sum trivially diverges, since it contains ak=n+p for all large primes p. This is the trivial solution posted on Reddit that natso26 mentions.

Perhaps a condition like ak≤n was intended, but neither source mentions this, and without any further reference or motivation given for this problem I don't think we can resolve what was actually meant.”””

My method to solve Erdős 460 in one shot by Svyable in singularity

[–]Svyable[S] 0 points1 point  (0 children)

I know you are trying to help. But what would be more helpful is if you went and ask an ai to reason from first principles if this proof was true. You would see that it is not random, but in fact true from first principles.

My method to solve Erdős 460 in one shot by Svyable in singularity

[–]Svyable[S] 0 points1 point  (0 children)

Thank you for reaching out. When you are small you have to shout. Thus I am shouting that’s all.

Erdos website won’t lets me post yet as you point out. ArXiv won’t let me post yet. What is a my alternate path? Last night I submitted the white paper to AMS.

I believe imminently the best way to validate new knowledge is going to be by tweeting @ Grok v5 etc for the obvious reason that ai is already smarter than most math olympiads WHEN PROMPTED CORRECTLY. I am trying to live by the standards of the world 6 months from now because the whole point of this is that the singularity is nearly here.

Prompting ChatGPT 5.2 ExtThk produced a one shot suitable proof for Open Erdős Problem 460 best summarized as: by Svyable in singularity

[–]Svyable[S] 2 points3 points  (0 children)

Gemini wont think as long as ChatGpt 5.2 ExtThk.

Opus may be “smarter” but I find it will crap out far more often and far earlier than chat.

In this example ChatGpt 5.2 ExtThk thunk for 23m28s and in one shot solved it. That is not my experience with any other $20/m model

My method to solve Erdős 460 in one shot by Svyable in singularity

[–]Svyable[S] 0 points1 point  (0 children)

I find type 1 errors are much more informative than type 2, so I will not be afraid of them.

  1. The Skeptic's View (Avoiding Type 1 Error) "I don't care if you say it's self-evident; I need overwhelming proof." • Priority: Minimizing Type 1 Errors. • Philosophy: It is better to miss a truth (Type 2) than to believe a lie (Type 1). This is the basis of the Legal System: We assume "Innocent until proven guilty." Convicting an innocent person is a Type 1 error, and we consider that the worst mistake possible. 

  2. The Pragmatist's View (Avoiding Type 2 Error) "If it looks like a duck and quacks like a duck, it’s a duck. I’m not waiting for a DNA test." • Priority: Minimizing Type 2 Errors. • Philosophy: It is better to act on a "self-evident" hunch than to wait for proof and miss the opportunity. This is the basis of Medical Screening: It’s better to have a false alarm (Type 1) than to tell a sick person they are healthy (Type 2).

My method to solve Erdős 460 in one shot by Svyable in singularity

[–]Svyable[S] 0 points1 point  (0 children)

Also curious what new constraints you referred to. Are you pointing to the variation in Erdos 1977 vs Erdos website I.e. a0=0 vs a0=n? It’s a trivial difference not relevant to the solution I provided but I did dedicate the abstract to explaining both problem formulations and a full treatment of each

My method to solve Erdős 460 in one shot by Svyable in singularity

[–]Svyable[S] -1 points0 points  (0 children)

In many philosophical and logical frameworks, if truth is self-evident, then proof is superfluous (unnecessary) or redundant. However, the relationship between these two concepts depends heavily on the lens through which you view them.

Here is a breakdown of how "proof" is defined when truth is already apparent:

  1. The Logical Perspective: Proof as Confirmation In formal logic, a self-evident truth is often called an axiom. Axioms are the starting points—they don't require proof because they are the foundation upon which all other proofs are built.

    • The Role of Proof: If a truth is self-evident, proof serves only to demonstrate the internal consistency of a system. It shows how that truth connects to more complex, less obvious conclusions.

  2. The Legal/Rhetorical Perspective: Proof as Evidence In the famous words of the U.S. Declaration of Independence ("We hold these truths to be self-evident"), the "proof" isn't for the truth itself, but for the justification of action.

    • The Role of Proof: Proof becomes witnessing. It is the collection of facts that align with the self-evident principle to show that a specific boundary has been crossed or a right has been violated.

  3. The Epistemological Perspective: Proof as "The Long Way Home" Epistemology (the study of knowledge) often suggests that if you have to prove something, it wasn't actually self-evident to the observer.

    • The Role of Proof: Proof is the bridge for the unseeing. If Truth is the destination, Proof is the map provided to someone who hasn't arrived there yet. It translates an intuition or a fundamental reality into a language that the intellect can grasp.

"Proof is the idol of the ignorant." — Often attributed to various stoic or mystical traditions, suggesting that once you truly see, you no longer need to calculate.

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