Reused writeup on Math-5813 (gpt pro 5.2)

Erdős Problem 675 --- collection record

PROVENANCE AND STATUS

Official problem: https://www.erdosproblems.com/675

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_5.2/675.tex.

REUSED SOURCE 1

Original attribution: gpt_pro_5.2. Original file: attacks/erdos/gpt_pro_5.2/675.tex.

1) FORMAL RESTATEMENT

A set $A\subset\mathbb N$ (here $\mathbb N=\{1,2,3,\dots\}$) has the translation property if for every integer $n\ge 1$ there exists an integer $t_n\ge 1$ such that for all integers $a$ with $1\le a\le n$, \[ a\in A\ \Longleftrightarrow\ a+t_n\in A. \] Equivalently, the membership pattern of $A$ on $\{1,2,\dots,n\}$ repeats on the translated block $\{1+t_n,2+t_n,\dots,n+t_n\}$.

Questions posed:

$Q1$ Does the set $A_{2\square}:=\{m\in\mathbb N: m=x^2+y^2\text{ for some integers }x,y\}$ have the translation property?

$Q2$ If the primes are partitioned into $P\sqcup Q$ with each part containing $\gg x/\log x$ primes $\le x$ for all large $x$, does the set \[ A_P:=\{m\in\mathbb N: \text{every prime divisor of }m\text{ lies in }P\} \] have the translation property?

$Q3$ Let $A_{\mathrm{sqf}}$ be the set of squarefree integers. Let $t_n^*$ be the minimal positive integer $t$ witnessing the translation property for $n$, i.e. \[ t_n^*:=\min\{t\ge 1: \forall a\le n,\ a\in A_{\mathrm{sqf}}\iff a+t\in A_{\mathrm{sqf}}\}. \] How fast can $t_n^*$ grow? In particular, is it true that $t_n^*>\exp(n^c)$ for some constant $c>0$?

2) QUICK LITERATURE/CONTEXT CHECK

As stated in the problem text: - Elementary sieve theory implies $A_{\mathrm{sqf}}$ has the translation property. - A more general sufficient condition is claimed using Brun's sieve: if $B\subseteq\mathbb N$ is pairwise coprime with $\sum_{b<x}1/b=o(\log\log x)$, then the set of integers not divisible by any $b\in B$ has the translation property.

Note: the problem text contains a notational slip: it says $A=\{n: b\nmid n\ \text{for all }b\in A\}$; the minimal correction consistent with the preceding sentence is \[ A=\{n: b\nmid n\ \text{for all }b\in B\}. \]

I do not use any other external results here.

3) ATTACK PLAN

- For $Q3$, compute $t_n^*$ for small $n$ to get growth data, and prove structural lemmas showing what congruence conditions automatically preserve (non-)squarefreeness. - I do not make progress on $Q1$ or $Q2$ beyond restating them.

4) WORK

FAST REALITY CHECK (squarefree numbers)

I computed $t_n^*$ for $1\le n\le 50$ by brute force search over $t$ (checking the squarefree indicator on $[1,n]$ and $[1+t,n+t]$). The values are piecewise constant with jumps: \[ \begin{array}{c|l} n & t_n^*\\\hline 1,2 & 1\\ 3,4 & 4\\ 5 & 12\\ 6,7,8 & 28\\ 9,\dots,12 & 36\\ 13,\dots,24 & 180\\ 25,\dots,30 & 900\\ 31,\dots,49 & 2376\\ 50 & 44100 \end{array} \] In particular, $t_{50}^*=44100=(2\cdot 3\cdot 5\cdot 7)^2$.

Lemma 675.1 (equivalent "prefix recurrence" formulation)

Let $A\subset\mathbb N$ and define its indicator $\mathbf 1_A(m)\in\{0,1\}$. Then $A$ has the translation property if and only if for every $n\ge 1$ there exists $t_n\ge 1$ such that \[ (\mathbf 1_A(1),\dots,\mathbf 1_A(n))=(\mathbf 1_A(1+t_n),\dots,\mathbf 1_A(n+t_n)). \]

Proof. The definition says that for all $1\le a\le n$ we have $a\in A\iff a+t_n\in A$. This is equivalent to saying $\mathbf 1_A(a)=\mathbf 1_A(a+t_n)$ for all $a\le n$, which is exactly the componentwise equality of the two length-$n$ vectors displayed.

Lemma 675.2 (a sufficient congruence condition that preserves small square divisors)

Fix $n\ge 1$ and let \[ P_n:=\{\text{primes }p: p^2\le n\}. \] Let $t$ be a positive integer satisfying \[ t\equiv 0\pmod{p^2}\quad\text{for every }p\in P_n. \] Then for every integer $a$ with $1\le a\le n$:

(a) If $a$ is divisible by $p^2$ for some $p\in P_n$, then $a+t$ is divisible by the same $p^2$, hence $a+t$ is not squarefree.

(b) If $a$ is squarefree, then for every $p\in P_n$ we have $p^2\nmid (a+t)$.

Proof. Fix $p\in P_n$. The congruence condition says $t\equiv 0\pmod{p^2}$, hence \[ a+t\equiv a\pmod{p^2}. \] Therefore $p^2\mid a$ if and only if $p^2\mid (a+t)$.

(a) If $p^2\mid a$ for some $p\in P_n$, then $p^2\mid(a+t)$, so $a+t$ has a square prime factor and is not squarefree.

(b) If $a$ is squarefree then for every prime $p$ we have $p^2\nmid a$, in particular for every $p\in P_n$. By the equivalence just proved, $p^2\nmid(a+t)$ for all $p\in P_n$.

Remarks on what this does not control: even if $a$ is squarefree, the number $a+t$ could still be divisible by $q^2$ for some prime $q$ with $q^2>n$ (e.g. $a=13$ and $t=36$ gives $a+t=49=7^2$). Thus Lemma 675.2 alone cannot produce a full translation for $A_{\mathrm{sqf}}$.

5) VERIFICATION

- Lemma 675.1 is a direct rewriting; quantifiers match. - Lemma 675.2: the key equivalence $p^2\mid a\iff p^2\mid(a+t)$ uses $t\equiv 0\pmod{p^2}$ and is correct for all integers $a$. - Computation sanity: the failure example $13+36=49$ is consistent with the jump $t_{13}^*=180$.

6) FINAL

UNRESOLVED

(i) Strongest proved partial result:

I computed the exact minimal shifts $t_n^*$ for squarefree numbers up to $n=50$ (table above). I also proved that choosing $t$ divisible by all prime squares $p^2\le n$ guarantees that the shifted block preserves all square divisibility by such small squares (Lemma 675.2), which is part of what is needed for $A_{\mathrm{sqf}}$.

(ii) First gap (crisp):

For $A_{\mathrm{sqf}}$, after enforcing the small-square congruences of Lemma 675.2, one must still ensure that for every squarefree $a\le n$, the number $a+t$ is not divisible by any prime square $q^2>n$. I do not have a general construction or lower bound mechanism controlling these larger squares.

(iii) Top 3 next moves:

1. Use a sieve/CRT argument to choose $t$ so that simultaneously (a) $t\equiv 0\pmod{p^2}$ for $p^2\le n$ and (b) $a+t$ avoids $0\pmod{q^2}$ for all squarefree $a\le n$ and all primes $q$ with $q^2\le n+t$. 2. Extend the brute-force computation of $t_n^*$ beyond $n=50$ and factor $t_n^*$ to guess a structural formula or at least a growth model. 3. For $Q1$ and $Q2$, compute $t_n^*$ for these sets for small $n$ to see whether prefix repetition even appears plausible; then attempt to connect any observed constraints to the defining arithmetic structure.

(iv) Minimal counterexample structure:

To show a set $A$ fails the translation property, one would need some $n$ such that the length-$n$ prefix of $\mathbf 1_A$ never reappears at any later position. For $A_{\mathrm{sqf}}$ the computations up to $n=50$ show repeated reoccurrence with moderately sized shifts, so a minimal counterexample (if it existed) would require a first $n$ where the prefix becomes "unique" in the infinite squarefree indicator sequence.

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.