gpt pro 5.2 attempt on Math-5607

1) FORMAL RESTATEMENT

1.1. Definitions

For an integer $k\ge 2$, a positive integer $n$ is called $k$-full if \[ \forall\text{ primes }p:\ p\mid n\ \Rightarrow\ p^k\mid n. \] Thus: * $2$-full $\Leftrightarrow$ powerful/squareful. * $3$-full is sometimes called cubeful.

1.2. Ambiguity/misstatement in the provided problem text

The problem text asks:

“Are there any 2-full $n$ such that $n+1$ is 3-full?”

but then cites consecutive examples $(8,9)$ with “8 is 3-full and 9 is 2-full” and $(12167,12168)$ with “$12167=23^3$ and $12168=2^3\cdot 3^2\cdot 13^2$” (again: first is 3-full, second is 2-full).

So the examples match the reversed condition.

I will therefore separate:

* Literal statement: find $n$ such that $n$ is 2-full and $n+1$ is 3-full. * Corrected statement (minimal change consistent with examples): find $n$ such that $n$ is 3-full and $n+1$ is 2-full; and ask whether the only such pairs are $(8,9)$ and $(12167,12168)$.

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2) QUICK LITERATURE/CONTEXT CHECK

I only use what is explicitly stated in the problem text:

* The pair $(8,9)$ has 8 = 3-full and 9 = 2-full. * Another pair is $(12167,12168)$ with $12167=23^3$ and $12168=2^3\cdot 3^2\cdot 13^2$. * It is stated (via OEIS A060355) that there are no other examples for $n<10^{22}$.

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3) ATTACK PLAN

1. Clarify the intended (corrected) formulation. 2. Prove structural lemmas for 2-full and 3-full integers, especially congruence constraints for consecutive pairs. 3. Do a modest computational scan to look for examples in both the literal and corrected directions.

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4) WORK

Phase 1: FAST REALITY CHECK (computation)

I searched up to $n\le 2\cdot 10^6$ and found:

* Literal direction (2-full $n$, 3-full $n+1$): no examples. * Corrected direction (3-full $n$, 2-full $n+1$): exactly two examples, $(n,n+1)=(8,9)$ and $(12167,12168)$.

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Lemma 4.1 (Parity constraints for consecutive 2-full/3-full)

Let $m\ge 2$.

1. If $n$ is 2-full and even, then $4\mid n$. 2. If $n$ is 3-full and even, then $8\mid n$.

Consequently, in any consecutive pair $(n,n+1)$ with one 2-full and the other 3-full:

* If the 3-full number is even, it is $0\pmod 8$. * If the 2-full number is even, it is $0\pmod 4$.

Proof. 1. If $n$ is 2-full and even, then $2\mid n$. By definition of 2-full, this implies $2^2\mid n$, i.e. $4\mid n$. 2. If $n$ is 3-full and even, then $2\mid n$. By definition of 3-full, $2^3\mid n$, i.e. $8\mid n$.

The consequences are immediate specializations. $\square$

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Lemma 4.2 (2-full $\Leftrightarrow a^2 b^3$ with $b$ squarefree)

A positive integer $n$ is 2-full if and only if $n=a^2 b^3$ with $a,b\in\mathbb N$ and $b$ squarefree.

Proof. This is exactly Lemma 4.1 in Problem #364 above, specialized to “2-full = powerful.” $\square$

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Proposition 4.3 (Verification of the stated examples for the corrected statement)

The pairs $(8,9)$ and $(12167,12168)$ satisfy: first number 3-full, second number 2-full.

Proof.

* $8=2^3$ is 3-full (the only prime divisor is 2, occurring to exponent 3). $9=3^2$ is 2-full. * $12167=23^3$ is 3-full. $12168 = 2^3\cdot 3^2\cdot 13^2$ has prime exponents $3,2,2$, all $\ge 2$, hence is 2-full.

$\square$

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5) VERIFICATION

* Lemma 4.1 is a direct reading of the definition applied to the prime $2$. * Lemma 4.2 was proved fully in Problem #364 (Lemma 4.1 there). * Proposition 4.3 uses explicit factorizations given in the problem text. * Computation: the scan is a sanity check only and is not used for any asymptotic or global claim.

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6) FINAL

UNRESOLVED

(i) Strongest fully proved partial result obtained here.

* Basic necessary congruence constraints for any such consecutive pair (Lemma 4.1). * Explicit verification that $(8,9)$ and $(12167,12168)$ satisfy the corrected condition (Proposition 4.3). * Computationally, no examples of the literal condition and only those two examples of the corrected condition were found for $n\le 2\cdot 10^6$.

(ii) Exact first gap.

For the corrected statement suggested by the examples:

Prove or disprove that the only solutions to “$n$ is 3-full and $n+1$ is 2-full” are $(n,n+1)=(8,9)$ and $(12167,12168)$.

(iii) Top 3 next moves (concrete targets).

1. Parametrize 2-full numbers as $a^2 b^3$ (Lemma 4.2) and similarly find a convenient parametrization of 3-full numbers to reduce the corrected problem to an explicit Diophantine equation. 2. Use gcd constraints: $\gcd(n,n+1)=1$ forces disjoint prime sets, which is very restrictive when both sides have only high prime exponents. 3. Extend computation with a sieve for k-full numbers to search for additional solutions (if any) and look for modular patterns that could be proved.

(iv) What a minimal counterexample would likely look like.

If the corrected statement has more solutions, the next one would be a much larger pair of consecutive integers $n,n+1$ with disjoint prime supports, where $n$ is cubeful (every prime exponent $\ge 3$) and $n+1$ is squareful (every prime exponent $\ge 2$). By Lemma 4.1, the even member must be divisible by $8$ (if it is cubeful) or $4$ (if it is squareful), so any counterexample must respect these strong 2-adic constraints.