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ERDőS #148 · PARTIAL

Erdős problem #148 — live-page audit and exact finite computation

Date of audit: 2026-07-26 UTC.

Claim labels used throughout:

0. Mandatory live-page gate

(d) I loaded https://www.erdosproblems.com/148 and its discussion thread on 2026-07-26 through the Bright Data residential browser path, not datacenter curl. The page title was “148 | Erdős Problems” and the page said it was last edited on 27 September 2025.

(d) The live status was OPEN. It displayed 0 claimed proofs, “Currently working on this problem: None”, and “Interested in collaborating: None”. Therefore the mandatory stop condition did not apply.

Verbatim live statement

The LaTeX-source view at https://www.erdosproblems.com/latex/148 gave the following statement verbatim:

> Let $F(k)$ be the number of solutions to\[ 1= \frac{1}{n_1}+\cdots+\frac{1}{n_k},\]where $1\leq n_1<\cdots<n_k$ are distinct integers. Find good estimates for $F(k)$.

Live-page known result

(b) The page lists

\[ 2^{c^{k/\log k}}\leq F(k) \leq c_0^{(1/5+o(1))2^k}, \qquad c_0=1.26408\ldots , \]

where \(c>0\) is absolute and \(c_0\) is called the Vardi constant. It attributes the lower bound to Konyagin [Ko14] and the upper bound to Elsholtz–Planitzer [ElPl21]. This is treated as the authoritative current statement, as requested.

All live markers and comments inspected

(d) The other displayed markers were: “Likes this problem: Dogmachine”, “This problem looks difficult: Dogmachine”, “This problem looks tractable: None”, “The results on this problem could be formalisable: None”, and “I am working on formalising the results on this problem: None”. Formalised statement was “No”; related OEIS entries were A076393 and A006585.

(d) There were two comments, both read in the discussion view:

1. Quanyu Tang, 2025-09-05, says that equation (4) in Konyagin’s paper is false, and points out that Elsholtz’s arXiv:1606.02117 independently supplies the same order of lower bound.

2. Woett, 2025-12-29, reports an email response from Konyagin agreeing that the displayed equation was mistaken, supplies a corrected equation, reports that Lemma 1 needs the hypothesis that \(m\) is odd, and says the main result still holds.

The site itself warns that comments are user responsibility and are not verified. I therefore do not treat the reported email or the “main result still holds” sentence as a proof. The corrected local identity is proved independently below, and Elsholtz’s published theorem supplies an independent route to the order of the lower bound.

The corrected formula, recovered from the live MathJax element’s exact data-latex, is

\[ \begin{aligned} \frac1{2^{3^k}-2^{3^{k-1}}} &=\frac1{2^{3^k}} +\frac1{(2^{2\cdot3^{k-1}}-1)2^{3^k}}\\ &=\frac1{2^{3^k}} +\frac1{(2^{2\cdot3^{k-1}}-1) (2^{3^k}+2^{3^k-m})}\\ &\hspace{22mm} +\frac1{(2^{2\cdot3^{k-1}}-1) (2^{3^k}+2^{3^k+m})}. \end{aligned} \tag{C} \]

1. Primary-source literature audit

Original problem

(b) Erdős and Graham, Old and New Problems and Results in Combinatorial Number Theory (1980), printed page 32, define the sets of strictly increasing solutions and ask for asymptotic formulae or good inequalities for their cardinalities. I inspected the scanned primary source:

https://mathweb.ucsd.edu/~ronspubs/80_11_number_theory.pdf

The live statement is a clean modern restatement of that question.

Lower bound

(b) Konyagin’s primary paper exists and was inspected in full from MathNet:

Its Theorem 1 states, in the notation \(X_n\) for these distinct-denominator representations,

\[ |X_n|\geq \exp\!\left( \exp\!\left( \left(\frac{(\log 2)(\log 3)}3+o(1)\right) \frac{n}{\log n} \right) \right). \]

This implies the coarser live-page lower-bound form.

(d) The printed primary source really does contain the suspect equation. At \(k=2,m=1\), its literal reading has

\[ \text{LHS}=\frac1{504},\qquad \text{RHS}=\frac{113}{24960}, \]

whose difference is \(1333/524160\neq0\). Reading the likely intended exponent as \(3^k-m\) instead gives RHS \(1/256\), still not \(1/504\). The standalone checker recomputes both failures exactly.

(a) Formula (C), by contrast, is an identity for every integer \(0 \[ N=3^k,\quad A=2^N,\quad B=2^{3^{k-1}},\quad Q=2^{2\cdot3^{k-1}}-1,\quad C=2^{N-m},\quad D=2^{N+m}. \]

Then \(A=B(Q+1)\), hence

\[ \frac1{A-B}-\frac1A=\frac1{AQ}. \]

Also \(CD=A^2\), so direct cross-multiplication gives

\[ \frac1{A+C}+\frac1{A+D}=\frac1A. \]

Multiplying the latter identity by \(1/Q\) proves (C). Moreover

\[ A+C=C(2^m+1),\qquad A+D=A(2^m+1), \]

so the final two denominators in (C) are divisible by \(2^m+1\), exactly as the second comment says. Parity of \(m\) is not needed for this identity; the comment locates the oddness requirement in Konyagin’s Lemma 1.

(b) There is an independent published lower-bound route:

Corollary 1.2 proves, for sufficiently large odd \(k\), at least

\[ \exp\!\left(\exp\!\left(c\,\frac{k}{\log k}\right)\right) \]

representations with distinct odd denominators, for some \(c>0\).

(a) The elementary injection

\[ (n_1,\ldots,n_k)\longmapsto (n_1,\ldots,n_{k-1},n_k+1,n_k(n_k+1)) \tag{I} \]

is valid because

\[ \frac1{n_k} =\frac1{n_k+1}+\frac1{n_k(n_k+1)} \]

and both new denominators exceed \(n_k\). It is injective, since the old \(n_k\) is recovered as the penultimate denominator minus one. Thus \(F(k+1)\geq F(k)\). Applying (I) once when necessary extends Elsholtz’s odd-\(k\) lower bound to all sufficiently large \(k\), with the same \(k/\log k\) order. Consequently the order of the live lower bound does not depend on accepting the unverified email report.

Upper bound

(b) The cited primary paper exists and was inspected:

Its Theorem 2 supplies the improved uniform estimates for sums of four and more unit fractions, and Corollary 3 specializes the lifted estimate to \(f_k(1,1)\). Here \(f_k\) counts weakly increasing denominators, so it is automatically an upper bound for the strictly increasing solutions counted by \(F(k)\).

(b) The shifted Sylvester-sequence normalization in Corollary 3 is easy to misread. Bloom and Elsholtz’s later survey, “Egyptian Fractions” (2022), page 241, states the consequence in the live page’s normalization exactly as

\[ F(k)\leq c_0^{(1/5+o(1))2^k}, \qquad c_0=1.26408\ldots . \]

Survey PDF:

https://www.math.tugraz.at/~elsholtz/WWW/papers/bloom-elsholtz-naw5-2022-23-4-237.pdf

(d) The verifier independently iterates the Sylvester recurrence

\[ s_0=2,\qquad s_{j+1}=s_j^2-s_j+1, \]

and obtains

\[ s_{10}^{\,1/2^{11}}=1.264084735305\ldots , \]

confirming the digits of the Vardi constant quoted on the live page.

Search for later work

(d) Targeted searches of arXiv, publisher pages, MathNet, the authors’ publication pages, and citation/phrase searches through 2026-07-26 found no primary source claiming a better asymptotic bound for this specific \(F(k)\), and no exact term beyond the \(k=8\) value recorded in OEIS A006585. This is a search result, not a theorem that no such paper exists.

(d) Two superficially nearby items were checked and not used as advances on #148:

2. Exact finite computation

Exhaustive algorithm

(a) At any search node, let the residual fraction be reduced to \(a/b>0\), let \(p\) be the preceding denominator, and let \(t\) terms remain. If the next denominator is \(x\) and \(t\geq2\), positivity of the later terms gives

\[ \frac1x<\frac ab, \]

while all \(t\) remaining unit fractions are at most \(1/x\), giving

\[ \frac ab\leq\frac tx. \]

Therefore every possible next denominator lies in the finite, exact interval

\[ \max\!\left(p+1,\left\lfloor\frac ba\right\rfloor+1\right) \leq x\leq \left\lfloor\frac{tb}{a}\right\rfloor . \tag{1} \]

The DFS loops over precisely (1), updates \((a/b)-1/x\) with integer arithmetic, reduces by a gcd, and recurses.

(a) With one term left, the only possible denominator is \(x=b/a\), accepted exactly when it is an integer larger than \(p\).

(a) With two terms left, the checker uses the bijective factor-pair identity

\[ \frac ab=\frac1x+\frac1y \quad\Longleftrightarrow\quad (ax-b)(ay-b)=b^2. \tag{2} \]

For every positive divisor \(d \[ x=\frac{b+d}{a},\qquad y=\frac{b+b^2/d}{a}, \]

for integrality and \(p

(a) The reduced residual denominator \(b\) divides the lcm of the chosen prefix denominators. Hence all of its prime factors occur in that prefix. The program factors \(b\) only over this support and asserts that the leftover factor is 1; it then generates every divisor of \(b^2\).

(d) Every emitted tuple is separately checked by taking its integer lcm \(L\) and asserting

\[ \sum_i L/n_i=L. \]

The \(k=7\) tuples are stored in a Python set, and insertion asserts that no duplicate occurs.

(d) A second implementation ignores (2), descends through every denominator using (1), and reaches only the one-term test. It independently agrees in count, ordered SHA-256 digest, and full solution set through \(k=6\). The factor-pair method is required for the inexpensive \(k=7\) run.

The core search is:

lower = max(previous + 1, denominator // numerator + 1)
upper = (terms_left * denominator) // numerator
for x in range(lower, upper + 1):
    new_numerator = numerator * x - denominator
    new_denominator = denominator * x
    g = gcd(new_numerator, new_denominator)
    search(new_numerator // g, new_denominator // g, x, terms_left - 1)

(d) The complete dependency-free implementation, including factor generation, the independent DFS, tuple audit, source-identity checks, and command-line interface, is:

runs/erdos148_reverify.py

SHA-256 of that source:

a5e11223202200129154e1242f41fec3c78ff8df46647c6dd61b658a9df1fbb1

Certified table

(d) The from-scratch run produced:

| \(k\) | \(F(k)\) | largest final denominator seen |

|---:|---:|---:|

| 1 | 1 | 1 |

| 2 | 0 | — |

| 3 | 1 | 6 |

| 4 | 6 | 42 |

| 5 | 72 | 1806 |

| 6 | 2320 | 3263442 |

| 7 | 245765 | 10650056950806 |

These counts agree with OEIS A006585 through \(k=7\), but the OEIS values were not used to generate the tuples; they appear only as post-computation assertions.

(d) The exact \(k=7\) first-denominator distribution is

\[ \#\{n_1=2\}=244817,\qquad \#\{n_1=3\}=948,\qquad \#\{n_1\geq4\}=0. \]

The sum is \(245765\).

(d) Among the \(k=7\) solutions, the smallest possible final denominator is 18, attained by

\[ (3,4,9,10,12,15,18), \]

and the largest final denominator found is \(10650056950806\), attained by

\[ (2,3,7,43,1807,3263443,10650056950806). \]

These extremal statements are exhaustive computational statements for \(k=7\), not uniform theorems.

(d) The complete lexicographically emitted \(k=7\) tuple stream (ASCII comma-separated tuples with a newline after each) has SHA-256

c6f59bc8d4e783f2425dc68eaeacbd244bbef73a303f818fef99cb02cb6567c2.

Reproduction command and output

Run from the repository root:

python3 runs/erdos148_reverify.py

(d) On CPython 3.12.3, the final timed run used 6.61 wall seconds, 6.54 user seconds, and 65,396 KiB maximum resident memory. Its output was:

source arithmetic: printed Konyagin identity fails at (k,m)=(2,1); corrected identity/divisibility PASS for [(2, 1), (2, 3), (2, 5), (2, 7), (3, 1), (3, 3), (3, 5), (3, 7), (3, 9), (4, 1), (4, 3), (4, 5), (4, 7), (4, 9)]
source arithmetic: Vardi recurrence approximation = 1.264084735305
factor-pair k=1: 1 sha256=4355a46b19d348dc2f57c046f8ef63d4538ebb936000f3c9ee954a27460dd865 max_last=1
factor-pair k=2: 0 sha256=e3b0c44298fc1c149afbf4c8996fb92427ae41e4649b934ca495991b7852b855 max_last=None
factor-pair k=3: 1 sha256=44a6d091e3cdb4a68e6f150bb970f60ab75f5fbc3894fefb764e63356f7a53ba max_last=6
factor-pair k=4: 6 sha256=39672b6b0b44b319c5422f1dabd5c688d4455481ebc06dc18dd598465d816d9e max_last=42
factor-pair k=5: 72 sha256=2e17ce836464201ce19704908ab14b2e86544ba1d6078d64d335ac3a1cb281a6 max_last=1806
factor-pair k=6: 2320 sha256=5b181f001414f2b605094dfed50a2fe64219f48df5bb72c44f47712435d0a891 max_last=3263442
factor-pair k=7: 245765 sha256=c6f59bc8d4e783f2425dc68eaeacbd244bbef73a303f818fef99cb02cb6567c2 max_last=10650056950806
direct-DFS k=1: 1 MATCH
direct-DFS k=2: 0 MATCH
direct-DFS k=3: 1 MATCH
direct-DFS k=4: 6 MATCH
direct-DFS k=5: 72 MATCH
direct-DFS k=6: 2320 MATCH
table F(1..7) = [1, 0, 1, 6, 72, 2320, 245765]
k=7 by first denominator = {2: 244817, 3: 948}
k=7 minimum last denominator = 18
k=7 minimizing tuple = (3, 4, 9, 10, 12, 15, 18)
k=7 maximum last denominator = 10650056950806
k=7 maximizing tuple = (2, 3, 7, 43, 1807, 3263443, 10650056950806)
k=7 tuple-set cardinality = 245765
k=7 ordered SHA-256 = c6f59bc8d4e783f2425dc68eaeacbd244bbef73a303f818fef99cb02cb6567c2
elapsed_seconds = 6.434410
ALL CHECKS PASSED

3. What this does and does not resolve

(a) Taking two logarithms of the live bounds leaves

\[ \Omega\!\left(\frac{k}{\log k}\right) \leq \log\log F(k) \leq O(k). \]

Thus the principal asymptotic gap is a factor of \(\log k\) on the \(\log\log F(k)\) scale. The exact \(k\leq7\) table does not shrink that uniform gap.

(c) On the lower-bound side, the Konyagin/Elsholtz mechanism obtains many choices from the divisors or primitive prime divisors of a highly composite exponent. The maximal-order divisor estimate supplies only \(\exp(\Theta(k/\log k))\) independently decodable choices within an \(O(k)\)-term budget. A concrete missing ingredient for an \(\exp(\exp(\Omega(k)))\) lower bound would be a replacement gadget that produces \(\exp(\Omega(k))\) independently recoverable choices using only \(O(k)\) distinct unit fractions.

(c) On the upper-bound side, Elsholtz–Planitzer obtain a nontrivial uniform estimate for a fixed four-term tail and lift it through the preceding denominators. Any fixed-tail lifting continues to yield an exponent proportional to \(2^k\). A concrete missing theorem for approaching the lower scale is a uniform \(r\)-unit-fraction bound with \(r=r(k)\to\infty\), strong enough under lifting that the resulting exponent in the outer exponential loses a factor comparable to \(\log k\). No such growing-tail parametrization was found in the searched literature.

(d) OEIS records \(F(8)=151182379\), but this run did not recompute it. The present verifier intentionally audits every solution, so an \(F(8)\) replay would require about 615 times as many callbacks as \(k=7\).

(c) Linear extrapolation from the measured run gives roughly 1.1 core-hours merely for the callback work and roughly 30 GiB if all \(k=8\) tuples are retained for duplicate detection; allowing for larger integer arithmetic, a realistic budget is 1–3 core-hours and at least 30 GiB for the same audit style. That exceeds the allowed few CPU-minutes and would only reproduce an already recorded term, so it was not run. A useful next exact advance would be \(F(9)\), but a tuple-by-tuple method is already bounded below by the \(F(8)\) workload via injection (I); it needs an aggregated three- or four-term counting routine before a credible cost can be given.

(d) Net result: the open asymptotic problem is not solved. The verified contribution here is an exact, reproducible \(k\leq7\) enumeration with independent smaller-case cross-checks, an additional audit distribution/certificate for \(k=7\), and an elementary verification of the correction appearing in the live discussion.

PARTIAL: exact exhaustive recomputation gives F(1..7) = 1,0,1,6,72,2320,245765 with a no-duplicate k=7 certificate; the corrected Konyagin identity is proved, but the asymptotic log-factor gap remains open.

This is the AI working report, labelled by outcome — not an independently verified claim unless marked PROVED. ← ledger