REVIEW
Unit fractions with semiprime denominators: an elementary proof of Erd\H{o}s Problem #306
Number Theory · Submitted 26 Sep 2026 · 9 references
Pith's verdict: A sound, genuinely elementary second proof of Erdős #306 with weaker prime inputs and a cleaner two-scale graph construction; the theorem predates it via Tang's Lean work, but the method and the honest write-up deserve a serious referee.
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Referee report
The paper
The paper claims
This paper gives an elementary proof of Erdős Problem #306: every positive rational whose reduced denominator is squarefree is a finite sum of distinct reciprocals of products of two distinct primes.
The reading
What carries the argument
The load-bearing object is a single complete bipartite graph between a small side $V=\{2\}\cup B\cup W$ ($B$ the primes of $b$, $W$ the primes in $(y^2,2y^2]$) and a large side $U$ of primes in $(y^8,Y]$, with $Y$ tuned so that half the total reciprocal mass $\mu(Y)$ sits within $H_V/(2y^8)$ below the target $\tau$. Each edge $\{v,u\}$ is the semiprime $vu$, and a subgraph is a set of distinct semiprime denominators. Counting subgraphs whose reciprocal sum matches $\tau$ modulo 1 is done by finite Fourier inversion: writing $h\mapsto(\xi,\zeta)$ through the Chinese remainder theorem, the Fourier coefficient $|\varphi(h)|$ becomes the product of entries of a $V\times U$ table whose $(v,u)$ entry is $|\cos\pi\varphi_{vu}|$. Frequencies are sorted by whether the row labels $\xi$ are coherent (all congruent to one small integer $M$ with $|M|\le y^7$) and whether column labels match $M$, giving three cases. Case I needs only positivity since $|\beta|=|\mu-\tau|\le H_V/(2y^8)$; case II collapses because a mismatched column loses weight at most $\varepsilon=\exp(-y^2/(2\cdot10^7\log^3 y))$ by a divisor count (Lemma 8.2) and the separation lemma (Lemma 8.3); case III loses $\exp(-\tau y^4/(64H_V))$ per column by the no-wrap-around lemma (Lemma 8.4), which beats the at most $2b\cdot16^{y^2}$ row patterns.
Extended reading notes
Core claim
The central claim, stated as Theorem 1.1, is that for any positive rational $a/b$ in lowest terms with $b$ squarefree there are distinct integers $1<n_1<\cdots<n_k$, each a product of two distinct primes, with $a/b=\sum_i 1/n_i$. The argument first reduces to small targets $\tau=a/b\le 1/400$, then builds one complete bipartite graph: the small side $V=\{2\}\cup B\cup W$, where $B$ holds the primes of $b$ and $W$ the primes in $(y^2,2y^2]$, and the large side $U$ the primes in $(y^8,Y]$ with $Y$ tuned so that half the total reciprocal mass of the graph sits just below $\tau$. Any subgraph then has mass at most $1/2$, so if its reciprocal sum is congruent to $\tau$ modulo 1 it is exactly $\tau$. Counting subgraphs as a finite Fourier sum, the frequencies split into three cases governed by a $V\times U$ table: the small integer frequencies give a positive main term of size $1/L$, and the minor arcs contribute at most $1/(2L)$ by a divisor-counting separation lemma and a no-wrap-around form of the Chinese remainder theorem. Because $y$ can be taken arbitrarily large, several copies of one small target can be padded into disjoint prime intervals, so $q$ copies of $a/(bq)$ sum to $a/b$.
Load-bearing premise
The proof stands on a single external, unproved-in-paper input: Ramanujan's inequality $\theta(x)-\theta(x/2)>x/6-3\sqrt{x}$ for $x>300$, cited from [7], which yields the large-side mass bound (Corollary 2.2, $\sum_{y^8<p\le y^9}1/p\ge 1/50$) and the test-prime count $|W|\ge y^2/(8\log y)$; if this bound failed, the tuning lemma, the separation arguments, and all minor-arc estimates would collapse.
Editorial extensions
If this is right
- Every positive rational with a squarefree reduced denominator, however small, admits an Egyptian-fraction representation using only distinct semiprime denominators; Theorem 1.1 subsumes the earlier threshold result of [5].
- Because $y$ can be chosen arbitrarily large, one representation can be padded into disjoint prime intervals $(y_i^8,y_i^9]$, so several copies of a small target combine without collisions; this is exactly how the general case $a/b$ is recovered from small targets.
- The method is parameter-flexible: with the small side at scale $y^2$ and the large side in $(y^{2k},y^{2k+2\lambda}]$, the same argument works for any fixed $k>3$ and $0<\lambda<1$ after adjusting the constants.
- The proof is supplied in machine-checkable form: Lean 4 verifies every step except the cited Ramanujan inequality, which is the single declared axiom of the formal theorem.
- All thresholds are effective and all constants explicit, so the theorem holds for every such rational without any asymptotic exceptions.
Reading between the lines
- The two-scale bipartite design suggests a template for the $k$-prime analogue: a $k$-partite graph between $k$ well-separated prime scales might carry the same Fourier counting for reciprocal sums of products of $k$ primes, extending the three-prime direction of Butler, Erdős and Graham.
- The no-wrap-around form of the Chinese remainder theorem (Lemma 8.4), where bounded phase numerators force numeric equality from congruence, is a general device that may transfer to other subset-sum congruence counts on prime graphs.
- Because the sole external axiom is a classical inequality that already implies Bertrand's postulate, the theorem is effectively unconditional in practice; an elementary verification of the bound on $\theta(x)-\theta(x/2)$ would make the machine-checked theorem fully axiom-free.
- The structure of tuned endpoint, positivity-only main term, and minor arcs handled by divisor counting may apply to counting problems where a target is reached modulo one under a bounded total mass.
Original abstract
We give an elementary proof that every positive rational number $a/b$ with $b$ squarefree is a finite sum of distinct unit fractions $1/n$, where each $n$ is a product of two distinct primes (Erd\H{o}s Problem #306). After a reduction to small targets, we take a single complete bipartite graph between the primes in $(y^2,2y^2]$, together with $2$ and the primes of $b$, and a tuned initial segment of the primes in $(y^8,y^9]$, and show that some subgraph has reciprocal sum congruent to $a/b$ modulo $1$; the small total mass then forces equality. Writing the number of such subgraphs as a finite Fourier sum, we sort the frequencies into three cases using a table indexed by the two sides of the graph. The small integer frequencies give a positive main term, and all other frequencies are negligible by a divisor-counting argument and a no-wrap-around form of the Chinese remainder theorem. The only inputs about primes are Chebyshev-type bounds. The circle-method framework comes from Tang's Lean development, which gave the first proof; our construction removes its anchor-synchronisation step. The proof has been formalised in Lean 4, apart from a cited inequality of Ramanujan. This work is a human-AI collaboration: AI tools contributed substantially to the construction, the experiments and the writing.
The compiled record
The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
The abstract, sentence by sentence
We give an elementary proof that every positive rational number $a/b$ with $b$ squarefree is a finite sum of distinct unit fractions $1/n$, where each $n$ is a product of two distinct primes (Erd\H{o}s Problem #306). After a reduction to small targets, we take a single complete bipartite graph between the primes in $(y^2,2y^2]$, together with $2$ and the primes of $b$, and a tuned initial segment of the primes in $(y^8,y^9]$, and show that some subgraph has reciprocal sum congruent to $a/b$ modulo $1$; the small total mass then forces equality. Writing the number of such subgraphs as a finite Fourier sum, we sort the frequencies into three cases using a table indexed by the two sides of the graph. The small integer frequencies give a positive main term, and all other frequencies are negligible by a divisor-counting argument and a no-wrap-around form of the Chinese remainder theorem. The only inputs about primes are Chebyshev-type bounds. The circle-method framework comes from Tang's Lean development, which gave the first proof; our construction removes its anchor-synchronisation step. The proof has been formalised in Lean 4, apart from a cited inequality of Ramanujan. This work is a human-AI collaboration: AI tools contributed substantially to the construction, the experiments and the writing.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We give an elementary proof that every positive rational number $a/b$ with $b$ squarefree is a finite sum of distinct unit fractions $1/n$, where each $n$ is a product of two distinct primes (Erd\H{o}s Problem #306). | contribution 0.55 | 0.96 | 0.97 | asserted 1.00 |
| s1 | After a reduction to small targets, we take a single complete bipartite graph between the primes in $(y^2,2y^2]$, together with $2$ and the primes of $b$, and a tuned initial segment of the primes in $(y^8,y^9]$, and show that some subgraph has reciprocal sum congruent to $a/b$ modulo $1$; the small total mass then forces equality. | method 0.99 | 0.94 | 0.06 | asserted 1.00 |
| s2 | Writing the number of such subgraphs as a finite Fourier sum, we sort the frequencies into three cases using a table indexed by the two sides of the graph. | method 0.99 | 0.92 | 0.02 | asserted 0.95 |
| s3 | The small integer frequencies give a positive main term, and all other frequencies are negligible by a divisor-counting argument and a no-wrap-around form of the Chinese remainder theorem. | method 0.98 | 0.89 | 0.02 | asserted 1.00 |
| s4 | The only inputs about primes are Chebyshev-type bounds. | method 0.68 | 0.53 | 0.02 | asserted 1.00 |
| s5 | The circle-method framework comes from Tang's Lean development, which gave the first proof; our construction removes its anchor-synchronisation step. | background 0.62 | 0.61 | 0.02 | asserted 1.00 |
| s6 | The proof has been formalised in Lean 4, apart from a cited inequality of Ramanujan. | limitation 0.70 | 0.82 | 0.03 | asserted 1.00 |
| s7 | This work is a human-AI collaboration: AI tools contributed substantially to the construction, the experiments and the writing. | method 0.47 | 0.60 | 0.02 | asserted 1.00 |
What was examined, and what was not
- arXiv listing abstract v1
- full text
- figures and tables
- references and citing sentences
- relations to other papers
- P abstract
- S sections
- T body sentences
- C citations
- R cross-paper relations
References
Discussion
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