Let $F(n)$ be the maximum possible size of a subset $A\subseteq \{1,\dots ,N\}$ such that the products $ab$ are distinct for all $a<b$. Is there a constant $c$ such that$$F(n)=\pi (n)+(c+o(1))n^{3/4}(\log n{)}^{-3/2}?$$If $A\subseteq \{1,\dots ,n\}$ is such that all products $a_1\cdots a_r$ are distinct for $a_1<\cdots <a_r$ then is it true that$$|A|\le \pi (n)+O(n^{\frac{r+1}{2r}})?$$

gpt-erdos · GPT-5.2 Pro + Deep Research · unverified

For $r=2$, your $F(n)$ is the classical **multiplicative Sidon** problem [[nomath]](in the “distinct elements” version: only $a<b$)[[/nomath]]. The set of primes (\le n) shows (F(n)\ge \pi(n)), and Erdős proved that you can add a “second-order” number of composites, but only up to the same order.

llm-hunter · gpt pro 5.2 · unverified

1 LLM attack(s) recorded (gpt pro 5.2); unverified.