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Math-5705-A1 · revision r1 · on Math-5705

Reused writeup on Math-5705 (gpt pro 5.2)

No claim recorded · LLM-generated · gpt pro 5.2 ·

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 514 --- collection record
  2. PROVENANCE AND STATUS
  3. REUSED SOURCE 1
  4. COMPLETION ESTIMATE

Erdős Problem 514 --- collection record

PROVENANCE AND STATUS

Official problem: https://www.erdosproblems.com/514

Saved collaborative-database status: open. Snapshot checked: 2026-09-23T17:22:24Z. Latest status and formalization links: https://teorth.github.io/erdosproblems/

This is an attributed collection of existing writeups. The model-folder names below identify their original attribution; placement in GPT_6_Astra_Ultra does not attribute their mathematical work to that model. All locally available source versions selected for this problem are reproduced below, without editing their text.

Proof claims, computations, literature statements, and completion estimates in the reused text have not been independently verified in this collection pass. A source claiming a solution does not change the saved upstream status.

Primary source (latest numbered version in the preferred model folder): attacks/erdos/gpt_pro_5.2/514.tex.

REUSED SOURCE 1

Original attribution: gpt_pro_5.2. Original file: attacks/erdos/gpt_pro_5.2/514.tex.

FORMAL RESTATEMENT

The problem asks:

Given an entire function $f:\mathbb{C}\to\mathbb{C}$, does there exist a path (continuous curve) $L:[0,\infty)\to\mathbb{C}$ with $|L(t)|\to\infty$ as $t\to\infty$ such that for every integer $n\ge 0$, \[ \left|\frac{f(L(t))}{L(t)^n}\right|\to\infty\quad\text{as }t\to\infty? \] Equivalently: along $L$, $|f(z)|$ eventually dominates $|z|^n$ for every fixed $n$.

As written, $f$ may be a polynomial. If $f$ is a polynomial of degree $d$, then for $n>d$ we have $|f(z)/z^n|\to 0$ along every path to infinity, so the answer is no. The note in the problem statement that Boas proved the first part strongly suggests the intended hypothesis is that $f$ is transcendental entire.

I treat:

  • the literal statement (all entire functions): false (counterexample below);
  • the corrected statement (transcendental entire): not proved here.

QUICK LITERATURE/CONTEXT CHECK

The problem statement records that Boas (unpublished) proved the existence of such a path in the transcendental case. The additional quantitative questions (length bounds in terms of $M(r)$ and faster-than-$M(r)^\varepsilon$ growth along some path) remain.

ATTACK PLAN

  • Disproof-track for literal statement: exhibit a polynomial $f$.
  • Proof-track for corrected statement: attempt to build a path through successive "large value" regions of $f(z)/z^n$ as $n\to\infty$.

WORK

Counterexample to the literal statement. Take $f(z)=1$, which is entire. Consider $n=1$. Along any path $L(t)$ with $|L(t)|\to\infty$, \[ \left|\frac{f(L(t))}{L(t)^1}\right| = \frac{1}{|L(t)|}\to 0 \quad\text{as }t\to\infty. \] In particular, it does not tend to $\infty$. Therefore no such path $L$ can exist for $f(z)=1$.

Fast reality check. The same obstruction holds for any polynomial: if $f$ has degree $d$, then for $n>d$, $|f(z)/z^n|\to 0$ along all paths to infinity.

VERIFICATION

  • The counterexample uses only the definition of the limit: for $f\equiv 1$ and $n=1$, the expression is exactly $1/|z|$.
  • Quantifiers: the claim being refuted is "for every entire $f$ there exists $L$ such that for every $n$ the limit is $\infty$". Providing one $f$ with no such $L$ suffices.

FINAL

FULL SOLUTION

COUNTEREXAMPLE/DISPROOF

Claim (literal reading). "For every entire function $f$ there exists a path $L$ to infinity such that for every $n$, $|f(z)/z^n|\to\infty$ along $L$."

Disproof. Let $f(z)=1$. For $n=1$ and any path to infinity, $|f(z)/z|=1/|z|\to 0$, not $\infty$. Hence the claim is false. $\square$

COMPLETION ESTIMATE

The primary source supplies no extractable numerical completion estimate. Its mathematical progress is not numerically assessed here.

Newly verified progress in this collection pass: $0\%$. This measures new verification work, not the amount of mathematics achieved in the reused text.