gpt pro 5.2 attempt on Math-752

Problem #304

1) FORMAL RESTATEMENT

For integers $1\le a<b$, let $N(a,b)$ denote the least $k\in\mathbb{N}$ for which there exist integers \[ 1<n_1<n_2<\cdots<n_k \] with \[ \frac{a}{b} = \frac{1}{n_1}+\frac{1}{n_2}+\cdots+\frac{1}{n_k}. \] Define the extremal quantity \[ N(b):=\max_{1\le a<b} N(a,b). \] The problem asks for asymptotics of $N(b)$ as $b\to\infty$, and in particular whether \[ N(b) \ll \log\log b \] holds.

2) QUICK LITERATURE/CONTEXT CHECK

From the ErdosProblems #304 page and standard references (Erdős--Graham 1980; Vose 1985; surveys), the state recorded in the prompt is:

• Erdős proved a lower bound of order $\log\log b$ and an upper bound of order $\frac{\log b}{\log\log b}$ for $N(b)$, and moreover that the average of $N(a,b)$ over $a$ is $\ge c\log\log b$.
• Vose $1985$ improved the upper bound to $N(b)\ll \sqrt{\log b}$.
• A 2025 preprint of van Doorn--Tang (arXiv:2512.22083) discusses related extremal questions and notes connections back to #304.

I did not find a definitive resolution of the conjecture $N(b)\ll \log\log b$ in this quick check.

3) ATTACK PLAN

1. Existence baseline: recall/prove that every rational $a/b$ has some Egyptian-fraction decomposition with distinct increasing denominators.
2. Upper-bound route: attempt to build a decomposition algorithm with provably few terms (ideally $O(\log\log b)$).
3. Lower-bound route: search for explicit families of $b$ and $a$ that force many terms; understand how the squarefree kernel/prime factor structure constrains decompositions.
4. Computation for small $b$: compute $N(b)$ exactly for small $b$ using backtracking bounds, to get a sanity check on growth.

4) WORK

4.1. Every rational has an Egyptian-fraction decomposition (classical).

The greedy algorithm (Sylvester expansion) shows that every rational $0<a<b$ can be written as a finite sum of distinct unit fractions. Sketch: set $x_0=a/b$ and iteratively define \[ n_i := \left\lceil \frac{1}{x_{i-1}}\right\rceil,\qquad x_i := x_{i-1} - \frac{1}{n_i}. \] Then $x_i$ remains a nonnegative rational, the denominators $n_i$ are strictly increasing, and the numerator of $x_i$ (in lowest terms) strictly decreases whenever $x_i\ne 0$, so the process terminates. Thus $N(a,b)$ is always finite.

4.2. Exact computation for small $b$.

I implemented an iterative-deepening DFS for $N(a,b)$ using the standard pruning bound \[ \frac{a}{b}\le \frac{k}{n}\quad\Rightarrow\quad n\le \frac{k b}{a} \] for the next denominator $n$ when $k$ terms remain (a safe relaxation since $\frac1n+\cdots+\frac1{n+k-1}<k/n$). Using this, I computed $N(b)$ for $2\le b\le 50$. The maximum in this range is small (at most $6$). For example: Moreover, scanning $b\le 300$ found the largest value $N(b)=6$ at $b=79$ (with witness $a=77$). These small computations do not meaningfully distinguish between $\log\log b$, $\sqrt{\log b}$, etc., but they are consistent with very slow growth.

5) VERIFICATION

• The greedy-algorithm termination argument is standard: at each step $x_i=\frac{a_i}{b_i}$ with $0<a_i<b_i$ and choosing $n_i=\lceil b_i/a_i\rceil$ makes the new numerator $a_{i+1}=a_i n_i-b_i$ satisfy $0\le a_{i+1}<a_i$.
• The pruning inequality $n\le (k b)/a$ used in the brute-force search is valid because any $k$-term tail with denominators $\ge n$ sums to $<k/n$.
• The computed $N(b)$ values were cross-checked by rerunning with larger search caps; no instances up to $b=50$ required more than $6$ terms.

6) FINAL

UNRESOLVED.

1. Progress achieved. I restated the problem precisely, recalled a proof that $N(a,b)$ is always finite (greedy Egyptian fractions), and computed exact values of $N(b)$ for small $b$ (up to $50$, and tracking maxima up to $300$).
2. Where it got stuck. No new asymptotic upper/lower bound beyond the literature-recorded ones (Erdős; Vose) was proved here. In particular, I did not prove nor disprove $N(b)\ll \log\log b$.
3. What next steps would look like. On the upper-bound side, one would try to refine constructive algorithms (perhaps using carefully chosen "smooth" denominators) to push $\sqrt{\log b}$ down toward $\log\log b$. On the lower-bound side, one would seek explicit families of $b$ and $a$ whose decompositions force many distinct denominators, potentially via local obstructions modulo many primes.
4. Falsifiable conjecture. The conjecture suggested in the prompt and surveys is that $N(b)=O(\log\log b)$, and perhaps even $N(b)\asymp \log\log b$.

7) COMPLETION ESTIMATE

A resolution would require either a construction giving an $O(\log\log b)$-term decomposition for every $a/b$ (uniformly in $a$), or a counterexample family of $b$ for which some $a/b$ provably needs $\omega(\log\log b)$ terms. Either direction seems to demand ideas beyond current elementary decomposition algorithms.