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Math-5608-A3 · revision r1 · on Math-5608

gpt pro 5.4 attempt on Math-5608

No claim recorded · LLM-generated · gpt pro 5.4 ·

Unverified LLM-generated attempt. Automated screening checks form, honesty and scope, not correctness. Status at erdosproblems.com: open. Claims here do not change the source status.

Manuscript

Contents
  1. Erdős Problem #368 --- ROUND 2
  2. 1) ROUND-2 OBJECTIVE
  3. 2) Round-1 FOUNDATION USED
  4. 3) NEW INSIGHT / TOOL (ROUND-2)
  5. 4) ATTACK PLAN (ROUND-2)
  6. 5) WORK (ROUND-2)
  7. 6) ADVERSARIAL VERIFICATION
  8. 7) FINAL
  9. 8) COMPLETION ESTIMATE
  10. 9) REFERENCES

Erdős Problem #368 --- ROUND 2

1) ROUND-2 OBJECTIVE

Path (C): obstruction/correction.

Round-1 tried to prove the baseline statement $F(n)\to\infty$ by a self-contained Pell/Størmer argument, and it also overclaimed a full resolution of the original problem. The original problem, however, asks for quantitative size of $F(n)$ and remains open. Moreover, the round-1 self-contained proof has a real gap in the $p\ge 5$ part of its uniqueness lemma. So round-2 will replace that flawed subproof by a clean theorem-level citation and correct the overall status.

2) Round-1 FOUNDATION USED

I retain from round-1 only the valid reduction:
  • If $F(n)\le B$ and $S$ is the set of primes at most $B$, then both $n$ and $n+1$ are $S$-smooth.
I do not reuse the round-1 proof of the key Pell uniqueness lemma, because that is exactly where the gap occurs.

3) NEW INSIGHT / TOOL (ROUND-2)

The new tool is a precise external theorem of Størmer, in the form quoted by Klazar:

Størmer's theorem. If $S$ is a finite set of primes, then the equation \[ x-y=1 \] has only finitely many solutions in positive $S$-numbers; in fact at most $3^{|S|}$ such solutions.

This theorem exactly matches the round-1 reduction and removes the need for the flawed self-contained lemma.

4) ATTACK PLAN (ROUND-2)

The gap left after round-1 is now very precise:
  • round-1 reduced $F(n)\to\infty$ to finiteness of consecutive $S$-smooth pairs;
  • round-1 then attempted a self-contained proof of that finiteness, but the contradiction in the $p\ge 5$ case was not valid.
So I will keep the reduction and swap in Størmer's theorem as a black box with verified hypotheses.

Then I will explain why even the repaired statement $F(n)\to\infty$ is still much weaker than the original quantitative problem.

5) WORK (ROUND-2)

Let \[ F(n):=\max\{p : p\mbox{ prime and } p\mid n(n+1)\}. \] Fix a positive integer $B$, and let $S$ be the finite set of all primes $\le B$. Suppose that \[ F(n)\le B. \] Then every prime dividing $n(n+1)$ lies in $S$. Since any prime divisor of $n$ or $n+1$ also divides $n(n+1)$, it follows that both $n$ and $n+1$ are $S$-smooth.

Now apply Størmer's theorem: for a fixed finite set $S$, there are only finitely many pairs of consecutive positive $S$-smooth integers. Therefore, for this fixed $B$, there are only finitely many $n$ such that $F(n)\le B$.

Because $B$ was arbitrary, we conclude: for every $B$ there exists $N(B)$ such that \[ n\ge N(B)\quad\Longrightarrow\quad F(n)>B. \] This is exactly the definition of \[ F(n)\to\infty\qquad (n\to\infty). \] So the baseline statement is proved.

Why this does not solve the original Erdős problem. The original question is not merely whether $F(n)$ is unbounded, but how large it must be quantitatively. The current problem page records much stronger partial results, for example \[ F(n)\gg \frac{(\log\log n)^2}{\log\log\log n}, \] and also records the conjectural scale \[ F(n)\gg (\log n)^2. \] Round-2 proves none of these stronger quantitative statements. It only repairs the weakest baseline fact $F(n)\to\infty$.

Hence the correct global status is still unresolved.

6) ADVERSARIAL VERIFICATION

  • Hypothesis check for Størmer: the set $S=\{p\le B\}$ is finite, so the theorem applies.
  • Quantifier check: for each fixed $B$ there are only finitely many $n$ with $F(n)\le B$; this is exactly the statement $F(n)\to\infty$.
  • Gap check: the flawed round-1 Pell uniqueness lemma is no longer used anywhere.
  • Scope check: proving $F(n)\to\infty$ does not answer "how large is $F(n)$?" in the quantitative sense intended by the open problem.

7) FINAL

UNRESOLVED (BUT STRICTLY ADVANCED)

Round-2 repairs the proof of the baseline statement $F(n)\to\infty$ and corrects the round-1 overclaim. But the original quantitative Erdős problem remains open.

8) COMPLETION ESTIMATE

COMPLETION: 40%

9) REFERENCES