GPT 6 Astra Ultra attempt on Math-753
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Erdős Problem #306: independent continuation
Date: 23 September 2026. Source versions read in full:306.tex in gpt_pro_5.4.1) OBJECTIVE AND SOURCE AUDIT
The question concerns every positive rational with squarefree reduced denominator, using distinct products of two distinct primes. The supplied graph sum identity is correct but selects the numerator rather than representing an arbitrary prescribed one. I use the same graph to obtain necessary cancellation constraints on any putative representation.2) WORK: WHERE LEAVES ARE ALLOWED
Suppose \[ a/b=\sum_{\{p,q\}\in E(G)}\frac1{pq},\qquad (a,b)=1, \] where $G$ is a finite simple graph on its nonisolated prime vertices. Put $P=\prod_{p\in V(G)}p$. Necessarily $b\mid P$. Multiplying the equation by $P$ and reducing modulo a vertex prime $p\nmid b$ gives \[ \sum_{q\in N_G(p)}q^{-1}\equiv0\pmod p. \tag{1} \] Indeed, the left side $aP/b$ is divisible by $p$, as are all edge contributions not incident with $p$. Dividing the remaining congruence by $P/p$, which is invertible modulo $p$, proves $1$.In particular every leaf prime must divide $b$: a single inverse cannot vanish. Hence a representation of an integer has minimum graph degree at least two. A representation with prime reduced denominator also cannot have a forest as its support graph. Every nonempty finite forest has at least two leaves, whereas at most one vertex prime can divide such a denominator. Thus these representations must use at least one cycle, not merely a tree of independently introduced primes.
More generally every nonempty tree component requires at least two distinct prime factors of $b$ among its leaves. If the graph has $t$ tree components, then $2t\le\omega(b)$. This is useful before any large-integer factorization in a proposed construction.