gpt pro 5.4 attempt on Math-882

Round-2 continuation for Erdős Problems 633, 634, 635

Correction to the Round-1 draft. The previous draft interchanged the problem numbers. The correct numbering is:

• Problem 633: classify triangles that admit only square-cardinality congruent-triangle dissections.
• Problem 634: determine all $N$ for which some triangle can be dissected into $N$ congruent triangles.
• Problem 635: the divisibility-avoidance problem for subsets of $\{1,\dots,N\}$.

This Round-2 file keeps the valid mathematical progress from Round 1, corrects the numbering error, and then strictly advances each problem.

Erdős Problem 634. Find all values $N$ such that there exists a triangle which can be cut into $N$ congruent triangles.

1) ROUND-2 OBJECTIVE

Path (C): obstruction/correction. The best Round-2 move is to correct the incomplete family list from Round 1. I will show that the known set of attainable $N$ is substantially larger than the Round-1 writeup recorded, by adding (i) the triquadratic family and (ii) a recent theorem of Zhang implying that for every fixed integer $m\ge 1$, all sufficiently large numbers of the form $mn^2$ are attainable.

2) Round-1 FOUNDATION USED

I rely on the following Round-1 material.

1. Every square $n^2$ lies in the attainable set $\mathcal N$.
2. The classical additional families recorded on the problem page are \[ 2n^2,\qquad 3n^2,\qquad 6n^2,\qquad n^2+m^2. \]
3. Beeson showed that $7,11\notin\mathcal N$.

3) NEW INSIGHT / TOOL (ROUND-2)

Two genuinely new ingredients are added.

1. Triquadratic family. Beeson III supplies an additional explicit infinite family, governed by the equation $N=2K^2-M^2$ with a divisibility condition.
2. Zhang's square-multiple theorem. The Erdős problem page now records a theorem implying that for every fixed $m\ge 1$, all sufficiently large values of the form $mn^2$ belong to $\mathcal N$.

These were missing from the Round-1 draft.

4) ATTACK PLAN (ROUND-2)

The gaps after Round 1 were:

1. the known existence side of $\mathcal N$ was incomplete;
2. no attempt was made to extract any corollary from the recent theorem quoted on the problem page.

I will now repair both gaps, while keeping the status honest: the exact determination of $\mathcal N$ is still open.

5) WORK (ROUND-2)

5.1. Classical attainable families.

From the problem page we have \[ n^2,\qquad 2n^2,\qquad 3n^2,\qquad 6n^2,\qquad n^2+m^2\in \mathcal N. \] We also know that $7,11\notin\mathcal N$, and $19$ is still open.

5.2. New family omitted in Round 1: triquadratic values.

Beeson's Triangle Tiling III gives a complete solution for the family of large triangles with angles $(2\alpha,\beta,\alpha+\beta)$. The criterion is: \[ N\in \mathcal N \text{ via such a triangle} \quad\Longleftrightarrow\quad N=2K^2-M^2 \] for some positive integers $K,M$ with $K\mid M^2$ (equivalently $K\mid N$).

This is a real enlargement of the known family list. For example, \[ 28=2\cdot 4^2-2^2 \] with $4\mid 2^2$, so $28\in\mathcal N$. Moreover $28$ is not of any of the classical forms listed in §5.1:

• it is not a square;
• it is not $2n^2$, $3n^2$, or $6n^2$;
• it is not a sum of two squares, since a direct check gives \[ 28-0^2=28,\quad 28-1^2=27,\quad 28-2^2=24,\quad 28-3^2=19,\quad 28-4^2=12,\quad 28-5^2=3, \] and none of these is a square.

Thus Round 1 really was incomplete: $28$ is a new explicit attainable value.

5.3. A strong corollary of the recent theorem on the problem page.

The current problem page records the following theorem: for integers $a\ge b$, if \[ n\ge 3\left\lceil \frac{a^2+b^2+ab-a-b}{ab}\right\rceil, \] then $n^2ab\in\mathcal N$.

Fix now any integer $m\ge 1$ and choose $(a,b)=(m,1)$. Then \[ \frac{a^2+b^2+ab-a-b}{ab} = \frac{m^2+1+m-m-1}{m}=m, \] so the theorem yields the explicit corollary \[ mn^2\in\mathcal N\qquad\text{for every } n\ge 3m. \] This is a substantial strengthening of the Round-1 discussion. In particular:

For every fixed $m\ge 1$, infinitely many multiples of $m$ belong to $\mathcal N$, and in fact every sufficiently large square-multiple $mn^2$ does.

5.4. What remains open.

The exact set $\mathcal N$ is still not known. The small obstructions $7$ and $11$ remain valid, and $19$ is still specifically open. The recent theorem proves large new infinite families, but not a complete characterization.

6) ADVERSARIAL VERIFICATION

Checks on possible weak points:

1. Could $28$ already belong to the old list? No; the direct calculation in §5.2 rules out the sum-of-two-squares possibility, and the other classical forms are obviously impossible.
2. Did the Zhang corollary use the theorem correctly? Yes. Substituting $(a,b)=(m,1)$ gives the threshold $n\ge 3m$ exactly.
3. Did I overclaim a full classification? No. I only enlarge the proven existence side of $\mathcal N$ and keep the exact determination open.

7) FINAL

UNRESOLVED (BUT STRICTLY ADVANCED).

Round 2 corrects the existence side of Problem 634 in a mathematically substantive way. Beyond the classical families $n^2,2n^2,3n^2,6n^2,n^2+m^2$, there is also the triquadratic family $N=2K^2-M^2$ with $K\mid M^2$, giving new explicit values such as $28$, and the current problem page records a theorem implying that for every fixed $m\ge 1$, all $mn^2$ with $n\ge 3m$ are attainable. The exact description of $\mathcal N$ remains open.

8) COMPLETION ESTIMATE (MANDATORY)

COMPLETION: 60%

9) REFERENCES

1. Erdős Problems, Problem 634.
2. Michael Beeson, Triangle Tiling III: The case $3\alpha+2\beta=\pi$.
3. Yan X. Zhang theorem as quoted on the current Erdős Problems page for Problem 634.