Erdoes #269 - working report - Erdős Problem a Day

Erdős problem #269 — exact finite-prime enclosures and the carry obstruction

Accessed and computed on 2026-07-29 (UTC).

Claim labels

[a] elementary-rigorous: proved below from integer arithmetic or a stated elementary lemma.
[b] rigorous modulo the explicitly named published theorem/source.
[c] plausible/structural-unverified: a diagnosis or proposed missing theorem, not a proof.
[d] computational-only: an exact finite computation or an observed search/page result. No floating-point calculation is used in any certified result.

Step 0: mandatory live-page gate

[d] I fetched the live problem page and its discussion thread through the Bright Data browser path on 2026-07-29. The page displayed OPEN, 0 claimed proofs, Currently working on this problem: None, and Interested in collaborating: None. Thus the requested stop condition was not met.

[d] Verbatim live statement (live page, with its LaTeX view):

Let $P$ be a finite set of primes with $\lvert P\rvert \geq 2$ and let $\{a_1<a_2<\cdots\}=\{ n\in \mathbb{N} : \textrm{if }p\mid n\textrm{ then }p\in P\}$. Is the sum\[\sum_{n=1}^\infty \frac{1}{[a_1,\ldots,a_n]},\]where $[a_1,\ldots,a_n]$ is the lowest common multiple of $a_1,\ldots,a_n$, irrational?

[d] The live page lists these known results:

If $P$ is infinite, the sum is irrational; Erdős called this a “simple exercise” in his 1988 paper.
Erdős's 1 January 1973 letter says that he could prove irrationality after duplicate LCM summands are removed.
The page was last edited 28 December 2025 and marks the statement as formalised.

[d] All external-data markers read on the page were: likes = Steve_Fan; interested in collaborating = None; currently working = None; looks difficult = None; looks tractable = None; results could be formalisable = None; working on formalisation = None.

[d] I read all seven comments on the discussion page. The site explicitly warns that comments are not verified.

Steve Fan, 26 June 2026: gives a Hecke–Mahler argument claiming transcendence for $|P|=2$, and explicitly says it does not immediately generalise to $|P|\geq3$.
old-bielefelder, 26 June 2026: observes that the two-generator calculation also works for arbitrary coprime integers.
Steve Fan, 26 June 2026: suggests a further algebraic-generator generalisation when the logarithms are $\mathbb Q$-linearly independent.
Steve Fan, 11 October 2025: gives an edited proof for infinite $P$.
Thomas Bloom, 11 October 2025: points out that an earlier assertion $x_{n+1}\geq2x_n$ was false and says the proof can be amended by estimating the tail directly.
Steve Fan, 11 October 2025: acknowledges the issue and says it will be fixed; the currently displayed earlier comment is the edited version.
Dogmachine, 11 October 2025: gives only a heuristic “meta-conjecture” comment.

[d] None of these is a full finite-$P$ proof claim, and no comment claims the $|P|\geq3$ case. I therefore proceeded, concentrating on that unresolved range while not duplicating the posted two-prime argument.

Primary-source and literature check

[b] The original one-page Fibonacci Quarterly letter, printed in volume 12 (1974), p. 335, states the conjecture and says that the series over distinct LCM values is irrational. The fetched PDF has SHA-256 473e2c82935b014c535eb1e80c2ca6fdfaa1af9b602acae5baa23500b346e202.

[b] Erdős and Graham, Old and New Problems and Results in Combinatorial Number Theory (1980), p. 65, write that the sum is irrational for infinite $Q$ and ask what happens for finite $Q$ with more than one element. I OCR-checked the scanned p. 65 against the displayed page. The fetched PDF has SHA-256 0cbf0c32f0ab1e1c71db5121a88bac905bf976c4a6ab6bb6d7d9cf9ddd184ed3.

[b] Erdős, “On the irrationality of certain series: problems and results”, New Advances in Transcendence Theory (1988), 102–109, DOI 10.1017/CBO9780511897184.009, repeats on p. 106 that infinite $P$ is a simple exercise and says the result “probably remains true” for finite $P$ of size greater than one. The fetched PDF has SHA-256 b2bfc375d04b65332d6b8817633ff3968283a3f33c1f1ace366b03ac9fab8c88.

[d] Exact-phrase searches for the 1974/1980 formulation, searches for the 1988 title and DOI, and screening the citation list of that DOI found no primary paper claiming this exact finite-prime problem. This is a documented search miss, not a proof that no such paper exists.

[d] One initially relevant-looking citation was C. Badea, “The irrationality of certain infinite series”, Glasgow Math. J. 29 (1987), 221–228, DOI 10.1017/S0017089500006868. Its section 5 solves two adjacent Erdős–Graham questions about Fibonacci/Lucas reciprocal series, not the prime-smooth LCM question.

[b] The theorem invoked by the posted two-prime route is of the right published type: Bugeaud and Laurent, arXiv:2203.12901, prove transcendence of the one-slope Hecke–Mahler value

F_{\theta,\rho}(z_1,z_2) =\sum_{k_1\geq1}\sum_{1\leq k_2\leq\lfloor k_1\theta+\rho\rfloor} z_1^{k_1}z_2^{k_2}

at the stated algebraic points. Luca, Ouaknine, and Worrell, arXiv:2412.07908, likewise treat a one-index, one-floor Hecke–Mahler series. Neither paper states a theorem for the simultaneous three-index carry correlation isolated below.

Exact elementary reformulation

Let $P=\{p_1,\ldots,p_k\}$, $k\geq2$, and write

m(\mathbf i)=\prod_{s=1}^k p_s^{i_s}, \qquad \mathbf i=(i_1,\ldots,i_k)\in\mathbb Z_{\geq0}^k.

[a] Lemma 1 (LCM at a smooth number). For every $P$-smooth $m$,

L(m):=\operatorname{lcm}\{u:u\leq m,\ u\text{ is }P\text{-smooth}\} =\prod_{r=1}^k p_r^{\lfloor\log_{p_r}m\rfloor}.

[a] Proof. The exponent of $p_r$ in the LCM is at most $\lfloor\log_{p_r}m\rfloor$. Equality holds because the smooth number $p_r^{\lfloor\log_{p_r}m\rfloor}$ is at most $m$. $\square$

[a] Corollary 2 (lattice series).

S(P)=\sum_{\mathbf i\geq0} \prod_{r=1}^k p_r^{-\left\lfloor \sum_{s=1}^k i_s\frac{\log p_s}{\log p_r} \right\rfloor}. \tag{1}

This is just the original series with its smooth numbers uniquely indexed by exponent vectors.

Put $R=\prod_{p\in P}p$. For $x=\log_p m$,

p^{\lfloor x\rfloor}>p^{x-1}=m/p.

Multiplying this over $p\in P$ gives

\frac1{L(m)}<\frac{R}{m^k}. \tag{2}

[a] Equation (2) proves absolute convergence, since

\sum_{\mathbf i\geq0}m(\mathbf i)^{-k} =\prod_{p\in P}(1-p^{-k})^{-1}<\infty.

A clean reduction: unary Hecke factors plus bounded carries

For $r\ne s$, define

\alpha_{rs}=\frac{\log p_s}{\log p_r},\qquad e_{rs}(n)=\lfloor n\alpha_{rs}\rfloor,

and

c_r(\mathbf i) =\left\lfloor\sum_{s\ne r}i_s\alpha_{rs}\right\rfloor -\sum_{s\ne r}e_{rs}(i_s). \tag{3}

[a] The elementary floor inequality gives

0\leq c_r(\mathbf i)\leq k-2. \tag{4}

Define the one-index weight

u_s(n)=p_s^{-n}\prod_{r\ne s}p_r^{-e_{rs}(n)}. \tag{5}

Splitting every floor in (1) into its separate floors plus (3) gives the exact identity

\boxed{ S(P)= \sum_{\mathbf i\geq0} \left(\prod_{s=1}^k u_s(i_s)\right) \left(\prod_{r=1}^k p_r^{-c_r(\mathbf i)}\right). } \tag{6}

[a] When $k=2$, every sum in (3) has one summand, so every carry is zero. Hence (6) factorises into two one-dimensional sums. This is precisely the algebraic feature exploited by the posted two-prime Hecke–Mahler argument.

[a] At $k=3$, each carry is $0$ or $1$, and carries genuinely occur. For $P=(2,3,5)$ and $\mathbf i=(0,1,2)$,

\lfloor\log_2(3\cdot5^2)\rfloor =\lfloor\log_2 75\rfloor=6,

whereas

\lfloor\log_2 3\rfloor+\lfloor\log_2 25\rfloor=1+4=5.

Thus $c_2(0,1,2)=1$.

For $P=\{p,q,r\}$, (6) is explicitly

\sum_{i,j,\ell\geq0} u_p(i)u_q(j)u_r(\ell)\, p^{-c_p(j,\ell)} q^{-c_q(i,\ell)} r^{-c_r(i,j)}. \tag{7}

[a] Formula (7) isolates the first unresolved obstruction: the no-carry part factorises, but three pairwise carry indicators couple the three indices.

[c] Precise missing transcendence lemma. A theorem proving irrationality (or transcendence) of the algebraic specialisation (7), under the multiplicative independence of $p,q,r$, would settle the three-prime case. The one-slope Hecke–Mahler theorems located in the literature do not state this correlated three-index result. For larger $k$, the exact analogue is (6) with carries in $\{0,\ldots,k-2\}$.

Exact finite enclosures

For an integer $M\geq0$, let

B_M=\{\mathbf i:0\leq i_s\leq M\ (1\leq s\leq k)\},\qquad S_M=\sum_{\mathbf i\in B_M}\frac1{L(m(\mathbf i))}.

[a] Lemma 3 (rational tail bound).

\[ 0<S(P)-S_M<U_M, \]

where the completely explicit rational number $U_M$ is

\boxed{ U_M= R\prod_{p\in P}(1-p^{-k})^{-1} \left(1-\prod_{p\in P}(1-p^{-k(M+1)})\right). } \tag{8}

[a] Proof. Apply (2) termwise outside $B_M$. The sum of $m(\mathbf i)^{-k}$ over all exponent vectors is the product of infinite geometric series. Its sum over $B_M$ is the same product multiplied by $\prod_p(1-p^{-k(M+1)})$. Their difference gives (8). Strict positivity follows because the complement of the box is nonempty. $\square$

[a] The program computes $S_M$ exactly over a common denominator

D_M=\prod_{p\in P} p^{\left\lfloor\log_p\left(\prod_{q\in P}q^M\right)\right\rfloor}.

Every term denominator in the box divides $D_M$, so the accumulation uses only integer additions and divisions.

Turning an enclosure into a denominator certificate

[a] Lemma 4 (Farey certificate). Suppose adjacent reduced fractions $a/b<c/d$ satisfy $bc-ad=1$. Every reduced fraction $h/k$ strictly between them has $k\geq b+d$.

[a] Proof. Both $hb-ak$ and $ck-dh$ are positive integers. Therefore

k=b(ck-dh)+d(hb-ak)\geq b+d.

$\square$

[a] Starting from $0/1<1/0$, the Stern–Brocot search retains the side containing the exact interval $(S_M,S_M+U_M)$. When the first mediant lies inside the interval, its denominator $b+d$ is, by Lemma 4, the least denominator of any rational in the interval. Since $S(P)$ lies strictly inside that interval, if $S(P)$ is rational in lowest terms, its denominator is at least this certified value.

Verified table

[d] The following covers every three-element subset of $\{2,3,5,7\}$, plus the four-element set itself. All displayed decimal digits are common to both exact rational endpoints; they are not floating-point approximations.

| $P$ | $M$ | exact terms | certified prefix of $S(P)$ | tail $<$ | least rational denominator in enclosure | |---|---:|---:|---|---:|---:| | $\{2,3,5\}$ | 90 | 753,571 | 1.809319352389316409569174677757… | $10^{-80}$ | 2,796,897,642,395,098,419,607,080,373,568,204,520,761 | | $\{2,3,7\}$ | 90 | 753,571 | 1.862602898514942932932598254551… | $10^{-80}$ | 5,773,395,375,806,770,795,458,229,120,736,546,781,477 | | $\{2,5,7\}$ | 90 | 753,571 | 1.822848468204189682538622734772… | $10^{-80}$ | 22,548,839,134,169,911,508,549,574,177,506,283,197,741 | | $\{3,5,7\}$ | 70 | 357,911 | 1.420443139264278445672602609594… | $10^{-99}$ | 142,396,196,159,863,645,410,061,607,193,662,519,047,098,020,948,996 | | $\{2,3,5,7\}$ | 30 | 923,521 | 1.7900166541509557014361030428… | $10^{-34}$ | 297,665,393,213,835,364 |

[d] Thus, for example, if $S(\{2,3,5\})$ is rational in lowest terms, its denominator is at least

\[ 2796897642395098419607080373568204520761. \]

This is a finite denominator exclusion, not an irrationality proof.

Standalone re-verification

[d] The standard-library-only verifier is runs/erdos269_wavew040_verify.py. Run:

python runs/erdos269_wavew040_verify.py

[d] It completed successfully on this VM in 16.4 seconds and printed ALL EXACT CHECKS PASSED. It performs the following independent checks before accepting the table:

Generates sorted $P$-smooth integers up to 10,000, updates their LCM directly, and compares against Lemma 1.
Checks (6) term-by-term on small exponent boxes using exact integer logarithms; it verifies all carries vanish for $(2,3)$ and verifies the explicit nonzero $(2,3,5)$ carry.
Recomputes small box sums by two independent routes: direct Fraction addition and common-denominator integer accumulation.
Checks (2) exactly on finite exponent boxes.
Recomputes every large enclosure, verifies its Farey-parent determinant is $1$, verifies the mediant is strictly inside, and matches fixed numerators, denominators, decimal prefixes, and SHA-256 fingerprints.

[d] The exact lower-endpoint SHA-256 fingerprints, in table order, are:

0e101309ee984b2a5b34d20d9ccdd2a8613a56457599594c3c8534cfb179559e
28dc144891f6277b63b527ce6788a7a46ec9ebd71681485966c5537eb368cbfb
4e595e1aa0f550c571436651cfb9d909415b44c7ac000762fbbd78ed11364da0
7024680f5fdd09f218ec637b1742c09a4d4e1cc7d49d0e6c18a3de2baf34325d
0e50c57768b1763280d44ce7590ddc41464758df30522bf5821337468b3c5487

What this does and does not settle

[a] General fast-denominator criteria do not apply directly to the original summands: as $n$ advances, the LCM either stays fixed or is multiplied by one of the finitely many primes in $P$. Its successive growth ratio is therefore bounded by $\max P$, whereas the Brun/Badea-type criterion used in the nearby 1987 paper requires eventual essentially quadratic growth in the unit-numerator case.

[a] The exact reason the posted two-prime factorisation stops is not merely “more variables”: it is the nonzero bounded carry tensor (3). Formula (6) removes every other coupling.

[c] The sharp analytic wall is a transcendence/irrationality theorem for the interacting torus-coded series (7), or its $k$-variable form (6). The primary Hecke–Mahler results found in the search handle one slope/floor coding and do not supply this lemma.

[a] The computational wall is uniformity. Increasing $M$ gives narrower exact intervals and larger finite denominator exclusions, but a rational number with a sufficiently large fixed denominator is compatible with every finite run. Closing the problem computationally would require a proof that the least Farey denominator of these enclosures tends to infinity with $M$; that missing uniform statement is itself an irrationality criterion and was not inferred from the table.

[d] Brute-force cost is not the central obstacle. A box has $(M+1)^k$ terms: at three primes, $M=200$ is about $8.1$ million terms (roughly tens of seconds to a couple of core-minutes in this implementation), while at four primes $M=90$ is about $68.6$ million terms (several core-minutes). Such runs would only enlarge finite denominator bounds and would not provide the missing uniform step.

PARTIAL: Exact carry-factor reduction for all finite P, plus rigorous rational enclosures excluding denominators below 40–51 digits for all three-prime subsets of {2,3,5,7} (and an 18-digit bound for {2,3,5,7}); irrationality for |P|>=3 remains open because a correlated multi-index Hecke–Mahler theorem or an equivalent uniform Farey-denominator argument is missing.

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