gpt pro 5.2 attempt on Math-5975

1) FORMAL RESTATEMENT

Fix $N\ge 1$. For subsets $A,B\subseteq\{1,2,\dots,N\}$ define, for each integer $m\ge 1$, \[ r_{A,B}(m):=\#\{(a,b)\in A\times B: ab=m\}. \] Define \[ F(A,B):=\#\{m\in\mathbb{N}: r_{A,B}(m)=1\}. \] Question: estimate \[ M(N):=\max_{A,B\subseteq\{1,\dots,N\}} F(A,B)\] as $N\to\infty$.

2) QUICK LITERATURE/CONTEXT CHECK

The problem statement records bounds (attributed there to van Doorn): \[ (1+o(1))\frac{N^2}{\log N}\le M(N)\ll \frac{N^2}{(\log N)^\delta(\log\log N)^{3/2}} \quad\text{with }\delta\approx 0.086. \]

3) ATTACK PLAN

• Give an explicit construction of $A,B$ with many uniquely represented products and prove uniqueness directly.
• Give simple general identities/inequalities involving $F(A,B)$ (e.g. characterizing when $F=|A||B|$).
• Do a brute-force reality check for small $N$.

4) WORK

PHASE 1 — FAST REALITY CHECK (brute force for small $N$).

I brute-forced $M(N)$ for $2\le N\le 10$ by enumerating all pairs of subsets $(A,B)$ (using bitmasks). The exact maxima found were: \[ \begin{array}{c|ccccccccc} N&2&3&4&5&6&7&8&9&10\\\hline M(N)&2&4&7&10&13&18&22&29&32 \end{array} \] (Examples of maximizers exist; one example for $N=10$ is $A=\{1,2,3,4,5,6,7,8\}$ and $B=\{1,7,8,9,10\}$, which yields $F(A,B)=32$.)

Problem-specific lemmas (injectivity and a clean lower-bound construction).

Lemma 896.1 (Injectivity criterion). For any finite sets $A,B$ of positive integers, \[ F(A,B)=|A||B| \quad\Longleftrightarrow\quad \text{the map }A\times B\to\mathbb{N},\ (a,b)\mapsto ab\text{ is injective}. \]

Proof. We always have $F(A,B)\le |A||B|$ since each $m$ with $r_{A,B}(m)=1$ accounts for exactly one pair $(a,b)\in A\times B$.

If the map is injective, then every product $ab$ occurs for exactly one pair, so $r_{A,B}(m)\in\{0,1\}$ for all $m$ and $F(A,B)=|A||B|$.

Conversely, if $F(A,B)=|A||B|$, then every pair $(a,b)\in A\times B$ must produce an $m=ab$ with $r_{A,B}(m)=1$ (otherwise we would have strictly fewer than $|A||B|$ values with exactly one representation). Thus distinct pairs cannot have the same product, i.e. the map is injective. $\square$

Lemma 896.2 (Large-prime construction gives many unique products). Let \[A:=\{1,2,\dots,\lfloor N/2\rfloor\},\qquad B:=\{p\ \text{prime}: N/2<p\le N\}. \] Then every $m$ of the form $m=ap$ with $a\in A$ and $p\in B$ has exactly one representation $m=ab$ with $a\in A$, $b\in B$. In particular \[ F(A,B)=|A||B|=\lfloor N/2\rfloor\left(\pi(N)-\pi(N/2)\right). \]

Proof. Fix $a\in A$ and $p\in B$ and let $m=ap$. We show that if $m=a'p'$ with $a'\in A$ and $p'\in B$, then necessarily $p'=p$ and $a'=a$.

Since $p\mid m$ and $p'$ is prime, either $p'=p$ or $p\mid a'$. But $p> N/2$ and $a'\le \lfloor N/2\rfloor$, so $p\nmid a'$. Hence $p'=p$, and then $a'=m/p=a$. Therefore $r_{A,B}(m)=1$ for every $m$ in the image of $A\times B$. Moreover the same argument shows the map $(a,p)\mapsto ap$ is injective, so by Lemma 896.1 we have $F(A,B)=|A||B|$. $\square$

5) VERIFICATION

• Lemma 896.1: the only subtle direction is that $F=|A||B|$ forces every pair $(a,b)$ to map to a uniquely represented product. This follows because the total number of pairs is $|A||B|$, and a value $m$ with $r(m)=1$ accounts for exactly one pair.
• Lemma 896.2: the key inequality is $p>N/2\ge a'$, so $p\nmid a'$. This is correct.

6) FINAL

UNRESOLVED

(i) Strongest proved partial result. The explicit choice $A=\{1,\dots,\lfloor N/2\rfloor\}$ and $B=\{p\text{ prime}: N/2<p\le N\}$ yields $F(A,B)=\lfloor N/2\rfloor(\pi(N)-\pi(N/2))$ (Lemma 896.2), i.e. on the order of $N\cdot(\#\text{primes in }(N/2,N])$. Exact brute-force maxima for $N\le 10$ are $M(2)=2, M(3)=4,\dots,M(10)=32$.

(ii) First gap (crisp). Prove an asymptotic formula (or a tight order of magnitude) for $M(N)=\max_{A,B\subseteq[N]}F(A,B)$ as $N\to\infty$. In particular, determine whether $M(N)\sim cN^2/\log N$ for some constant $c>0$.

(iii) Top 3 next moves.

1. Improve the lower bound construction by allowing $B$ to contain integers with a guaranteed “unique large prime factor” beyond $N/2$, while enlarging $A$.
2. Develop an upper bound mechanism: bound the number of uniquely represented products by controlling collisions $ab=a'b'$ via divisor statistics or energy methods.
3. Computational: for moderate $N$ (say $N\le 18$) use ILP/SAT to find exact $M(N)$ and examine the structure of extremizers, to guess the right asymptotic constant and shape.

(iv) Minimal counterexample structure (if the conjectured order is wrong). If $M(N)$ were substantially larger than $N^2/\log N$, extremizers would need to avoid the ubiquitous multiplicative collisions in $[1,N]^2$; this would likely force $A$ and $B$ to concentrate on numbers with a “signature” prime factor in a narrow range, so that products carry an essentially unique large prime.