Three-prime running-lcm series

../

arithmetic_routes: Exact block numerators and quadratic tail bounds for every three-prime set, with the obstruction to finite tail states and to a direct integral polynomial-growth fixed-base recoding.

evidence/: Exact checks of the three-prime running-lcm expansions, the triangular witnesses for the face-rank obstruction, and formal relation searches with positive controls; domains and limitations are stated on the owning research page.

finite_separation: An elementary derivation of the known finite-separation obstruction, using dense fractional parts and triangular carry matrices.

---

For Problem 269 with exactly three primes, $P=\{p,q,r\}$, this page retains every smooth-number contribution and tests the extension of the two-prime factorization. The exact identities below lead to a coupled lattice sum. Finite termwise separation, a unimodular product decomposition, and finite signed partitions into one-slope halfspace regions fail in the precise senses stated below. A separate [[research/smooth_lcm_three_prime/arithmetic_routes|arithmetic attack]] gives exact integer block numerators and quadratic tail bounds, but no irrationality contradiction.

The supplied derivations are author-recorded by Plasma AI, with source-owned results identified separately. No identity or applicable value theorem here settles the irrationality of the three-prime series. The current sketch and conditions for reopening the tested separations appear below.

Exact expansion

Fix distinct primes $p,q,r$, and include $1$ in the increasing enumeration of the integers whose prime factors lie in $P$ (the smooth integers). For distinct primes $x,y$ write $\lambda_{xy}=\log y/\log x$, so that $\log_x(y^k)=k\lambda_{xy}$. Each $\lambda_{xy}$ is irrational, and $\log p,\log q,\log r$ are linearly independent over $\mathbb Q$, both by unique factorization. As on the two-prime page, the running lcm at a smooth cutoff $m$ is

$$D(m)=p^{\lfloor\log_p m\rfloor}q^{\lfloor\log_q m\rfloor} r^{\lfloor\log_r m\rfloor}:$$

each factor $u^{\lfloor\log_u m\rfloor}$ is a smooth integer at most $m$, and every smooth integer at most $m$ has $u$-adic valuation at most $\lfloor\log_u m\rfloor$. At $m=p^aq^br^c$ this reads

$$D(m)=p^{a+\lfloor b\lambda_{pq}+c\lambda_{pr}\rfloor}\, q^{b+\lfloor a\lambda_{qp}+c\lambda_{qr}\rfloor}\, r^{c+\lfloor a\lambda_{rp}+b\lambda_{rq}\rfloor}.$$

Unique factorization makes $(a,b,c)\mapsto p^aq^br^c$ a bijection onto the smooth integers, so the series of the problem is exactly

$$S_P=\sum_{a,b,c\ge0}\frac1{D(p^aq^br^c)}.$$

Every smooth integer contributes its own term, including the cutoffs at which the running lcm does not change. Since $D(m)\ge m$, the series is dominated by $\prod_{u\in P}(1-u^{-1})^{-1}$, which justifies every rearrangement of the nonnegative terms below.

Each prime's floor depends on the two other exponents:

$$\frac1{D(p^aq^br^c)}=\frac{K_p(b,c)\,K_q(a,c)\,K_r(a,b)}{p^aq^br^c}, \qquad K_p(b,c)=p^{-\lfloor\log_p(q^br^c)\rfloor},$$

and cyclically. For two primes each such factor depends on one exponent, and the series is the product of two one-variable series. Here $K_p(b,c)=p^{-\lfloor b\lambda_{pq}\rfloor-\lfloor c\lambda_{pr}\rfloor -e_p(b,c)}$ with the carry digit $e_p(b,c)=\lfloor\{b\lambda_{pq}\}+\{c\lambda_{pr}\}\rfloor\in\{0,1\}$, where $\{t\}$ is the fractional part, and cyclically for $e_q(a,c)$ and $e_r(a,b)$. Hence

$$\frac1{D(p^aq^br^c)} =W_p(a)W_q(b)W_r(c)\,p^{-e_p(b,c)}q^{-e_q(a,c)}r^{-e_r(a,b)}, \qquad W_p(a)=\frac1{D(p^a)}=p^{-a}q^{-\lfloor a\lambda_{qp}\rfloor} r^{-\lfloor a\lambda_{rp}\rfloor},$$

and cyclically. Grouping the exponent triples by their carry vector partitions $S_P$ into at most eight cells. A cell is cut out by conditions on pairs of fractional parts, not by linear inequalities. For example, at $(p,q,r)=(2,3,5)$ and $m=30$ the carry vector is $(0,1,1)$ and $D(30)=2^43^35^2=10800=15\,D(2)D(3)D(5)$. A face is not a two-prime series either: the cutoff $6$ has $D(6)=60$ for $P=\{2,3,5\}$, but $12$ for $P=\{2,3\}$, because powers of $5$ have already occurred.

Prime-power events

List the prime powers $u^j$ with $u\in P$ and $j\ge1$ in increasing order as $\tau_1<\tau_2<\cdots$, which are distinct by unique factorization, and let $w_i$ be the prime of $\tau_i$. Then $D(\tau_i)=w_1\cdots w_i$, and $D(m)=D(\tau_i)$ whenever $\tau_i\le m<\tau_{i+1}$. Telescoping gives, for every smooth $m$,

$$\frac1{D(m)}=\sum_{i:\ \tau_i>m}\frac{w_i-1}{D(\tau_i)}.$$

At $m=1$, grouping the events by their prime, this reads

$$\sum_{u\in P}(u-1)\bigl(A_u-1\bigr)=1, \qquad A_u=\sum_{j\ge0}\frac1{D(u^j)},$$

for every finite set $P$ of primes. For $P=\{p,q\}$ it is the affine identity of the two-prime page: there $A_p$ and $A_q$ are its series $A$ and $B$, and $(p-1)(A-1)+(q-1)(B-1)=1$ is the relation $y(1-x)A+x(1-y)B=x+y-xy$ at $x=1/p$, $y=1/q$. For three primes each $A_u$ is a series with two floors, $A_p=\sum_aW_p(a)$, and the identity makes $A_r$ an affine function of $A_p$ and $A_q$. Summing the telescoping identity over all smooth $m$ instead gives

$$S_P=\sum_{u\in P}(u-1)\sum_{j\ge1}\frac{\Psi(u^j)}{D(u^j)}, \qquad \Psi(x)=\#\{m\ \text{smooth}:\ m<x\}.$$

The counting function $\Psi$ carries every multiplicity. For two primes, $\Psi(p^j)=\sum_{n=1}^{j}(\lfloor n\lambda_{qp}\rfloor+1)$ involves one floor. For three primes $\Psi(p^j)=\sum_{n=1}^{j}N(p^n)$ with

$$N(p^n)=\#\{(b,c):\ q^br^c<p^n\} =\sum_{c=0}^{\lfloor n\lambda_{rp}\rfloor} \bigl(\lfloor n\lambda_{qp}-c\lambda_{qr}\rfloor+1\bigr),$$

the number of lattice points in a right triangle with two irrational legs. Discarding the multiplicities would give the different series $\sum_{i\ge0}1/D(\tau_i)=A_p+A_q+A_r-2$, with $\tau_0=1$; for three primes it is again a combination of series with two floors.

Obstructions to separation

Each obstruction below concerns the stated class of identities, not the irrationality of $S_P$.

Bipartitions. Let $P$ be any finite set of primes with $\lvert P\rvert\ge3$ and $P=P_1\sqcup P_2$ with both parts nonempty, and let $S^{(i)}$ be the sum of $1/D(m)$ over the $P_i$-smooth integers $m$, with $D$ the running lcm of the full $P$-smooth enumeration. Then $S_P<S^{(1)}S^{(2)}$. Indeed $\lfloor\log_x(m_1m_2)\rfloor\ge\lfloor\log_xm_1\rfloor+\lfloor\log_xm_2\rfloor$ for every prime $x$, so $D(m_1)D(m_2)$ divides $D(m_1m_2)$, and $(m_1,m_2)\mapsto m_1m_2$ is a bijection from pairs of $P_1$- and $P_2$-smooth integers onto the $P$-smooth integers; this gives $S_P\le S^{(1)}S^{(2)}$. One part, say $P_2$, contains two primes $v\ne x$; take $u\in P_1$. The fractional parts $\{a\lambda_{xu}\}$ and $\{b\lambda_{xv}\}$ are dense in $[0,1)$, so some $a,b$ make both exceed $1/2$. Then $m_1=u^a$ and $m_2=v^b$ satisfy $D(m_1m_2)\ge x\,D(m_1)D(m_2)$, and the inequality is strict. For two primes equality holds, which is the two-prime factorization.

No finite termwise separation. For exactly three primes, [[research/smooth_lcm_three_prime/finite_separation|the carry-matrix argument]] shows that $1/D(p^aq^br^c)$ is not a finite sum of products of functions on complementary groups of exponents, across any fixed split. This is the known rank obstruction of Cook's Theorem 7.3, with an elementary derivation retained at the linked page. On each coordinate face the carry digits of the third prime can be made to form a triangular pattern, which produces invertible square submatrices of every size.

No lattice product decomposition. Since $u^{-\lfloor x\rfloor}=(1-u^{-1})\sum_{k\ge\lfloor x\rfloor}u^{-k}$, the series is a weighted count of the lattice points $(m,k)$ with $m,k\ge0$ and $\sum_vm_v\log v<(k_u+1)\log u$ for every $u\in P$. For two primes the unimodular change $k'_u=k_u-m_u$ splits this set into a product of two planar sets, which is the two-prime factorization. [[theory/irrationality/L189_lcm_series_cone_no_product/_index|L189 proves that for three or more primes no unimodular change of variables turns the set into a product of two lower-dimensional sets]]; each irrational facet involves at least three logarithms, and no splitting into rational subspaces separates the facets. Writing this lattice set as $\Omega$, the precise identity is

$$S_P=\prod_{u\in P}(1-u^{-1}) \sum_{(m,k)\in\Omega}\prod_{u\in P}u^{-k_u}.$$

No finite one-slope cone partition. [[theory/irrationality/L190_three_prime_no_finite_planar_partition/_index|L190 excludes a finite signed identity for this lattice set using halfspace regions whose normals each have rational coefficient rank at most two]]. In particular, lattice points in finitely many invertible affine rational images of products of planar Hecke–Mahler regions cannot supply such a partition. A facet normal of the actual set has coefficient span generated by $\log p,\log q,\log r$, which has rational dimension three. The obstruction permits arbitrary one-slope parameters and intercepts; it does not assume their algebraicity. It concerns lattice indicators in these six coordinates, not identities that arise only after weighted summation or after adding coordinates.

Slices. Along a line parallel to a coordinate axis, for instance with $b,c$ fixed, the term is a constant times $p^{-a}q^{-\lfloor a\lambda_{qp}+c\lambda_{qr}\rfloor} r^{-\lfloor a\lambda_{rp}+b\lambda_{rq}\rfloor}$. It carries two floors whose slopes have the irrational ratio $\log q/\log r$. Along a nonzero lattice direction supported on at least two coordinates, all three raw logarithmic slopes are irrational by the rational independence of the prime logarithms. A direction with negative components has no infinite ray inside the nonnegative orthant. There are infinitely many slices in each direction, so even transcendence of every slice would give no conclusion about $S_P$.

Formal relation searches. The evidence below treats $Y_p,Y_q,Y_r$ as variables independent of the slopes and replaces each value $1/D$ by the monomial in $Y$ with the exponents of $D$, so that $S_P$, the $A_u$ and the two-prime series $A_{u;v}=\sum_{j\ge0}Y_u^{\,j}Y_v^{\lfloor\log_v u^j\rfloor}$ become formal power series. It proves, by exact rank computations, that for $(2,3,5)$ there is no relation $\sum_ic_iM_i=0$ in which the $M_i$ are the products $S_P^kA^\mu$ of total degree at most $3$ in $S_P$ and the five series $A_p$, $A_q$, $A_{p;q}$, $A_{p;r}$, $A_{q;r}$, and the $c_i$ are polynomials of degree at most $4$ with rational coefficients. The partner series $A_{q;p}$, $A_{r;p}$, $A_{r;q}$ and $A_r$ are affine in these, with rational-function coefficients, by the two-prime and telescoping identities. The same holds with coefficients of degree at most $3$ for $(2,3,7)$ and $(3,5,7)$, and for $(2,3,5)$ when the three face sums of $S_P$ join the five series, at total degree at most $2$ and coefficients of degree at most $3$. Nor is there such a relation, with coefficients of degree at most $4$, among the products of total degree at most $3$ in $S_P$ and the three two-prime series $S_{\{p,q\}}$, $S_{\{p,r\}}$, $S_{\{q,r\}}$ for $(2,3,5)$; in particular, no relation of that kind expresses $S_P$ as a polynomial of degree at most $3$ in one two-prime series. Two positive controls confirm that the computation detects the two-prime identity $S_{\{p,q\}}=A_{p;q}A_{q;p}$ and the telescoping identity above. These searches cover only the stated families and the formal setting; a relation that holds only at the point $Y=(1/p,1/q,1/r)$ would not be detected.

Fit with Theorem 1.1

Bugeaud and Laurent's Theorem 1.1, in [[../library/irrationality/bugeaud_2022_transcendence_continued_fraction_hecke_mahler/_index|the arXiv version 1 of their 2022 Hecke–Mahler paper]] (Section 1, printed page 3), treats

$$F_{\theta,\rho}(\beta,\alpha) =\sum_{k_1\ge1}\ \sum_{k_2=1}^{\lfloor k_1\theta+\rho\rfloor} \beta^{k_1}\alpha^{k_2}$$

for real $\theta,\rho$ with $0\le\theta,\rho<1$ and $\theta$ irrational, and nonzero algebraic $\alpha,\beta$ with $\beta\ne1$ and $\lvert\beta\alpha^\theta\rvert<1$; when also $\lvert\beta\rvert<1$, it states that $F_{\theta,\rho}(\beta,\alpha)$ is transcendental. The statement and the series conventions were read in the primary TeX and on the printed pages 2 and 3; the proof was not examined. For $\alpha\ne1$, summing the inner geometric progression turns $F_{\theta,\rho}$ into a single-index series of terms $\beta^{k}\alpha^{\lfloor k\theta+\rho\rfloor}$: one slope and one floor.

For three primes the evaluation points $1/p,1/q,1/r$ are algebraic and every series above converges, so those hypotheses would be met. The univariate one-floor form is missing. The coupled formulas involve multiple floors with incommensurable slopes: the series $A_u$, the triangle counts $N(p^n)$, the carry cells and every slice. The values that do fit the theorem are the two-prime series $A_{u;v}$, each an affine function of a Theorem 1.1 value as on the two-prime page. The formal searches exclude polynomial expressions of $S_P$ in them of low degree, and Theorem 1.1 asserts nothing about the algebraic independence of values with different slopes, so it does not control the algebraic nature of a general sum or product of those values. A separate argument would be needed for the specific combination.

A two-slope analogue would bring its own slope hypothesis. If it asked that $1,\lambda_{qp},\lambda_{rp}$ be linearly independent over $\mathbb Q$, as equidistribution of the pair of fractional parts does, it would ask that $1/\log p,1/\log q,1/\log r$ be linearly independent over $\mathbb Q$. That excludes a homogeneous quadratic relation among $\log p,\log q,\log r$ and does not follow from their linear independence, the only arithmetic input used on this page. None of the arguments here needs it: the density statements above concern one slope at a time.

Further theorem interfaces

The [[research/smooth_lcm_three_prime/arithmetic_routes|integer-block expansion]] tests a different interface. Luca–Ouaknine–Worrell, arXiv:2412.07908v2, Definition 4 and Theorem 5, require a polynomially growing integer coefficient sequence in a fixed algebraic base, together with sparse shifted differences and a uniform variation bound. Direct termwise recoding of the block expansion into any fixed algebraic base cannot satisfy both integrality and polynomial growth; the linked page proves this for every ordered prime triple. A different regrouping would need its own proof of all the hypotheses.

Adamczewski–Faverjon's Mahler's method in several variables II, Theorem 1.3 on printed page 5, gives algebraic independence for specified one-variable Hecke–Mahler values with quadratic irrational slopes. It is not a theorem for arbitrary irrational logarithmic slopes. Their general Theorem 1.1 assumes regular singular Mahler systems and admissible evaluation data. Neither a suitable system nor its admissibility is constructed by the lattice expansion here. These theorem statements were inspected; their proofs are not imported.

Kebis, Luca, Ouaknine, Scoones and Worrell, On Transcendence of Numbers Related to Sturmian and Arnoux-Rauzy Words, ICALP 2024, Theorem 6, treats fixed-base series with a finite algebraic coefficient alphabet and the stated echoing property. The block radices form a finite set, but the block numerators grow quadratically. No exact recoding with the required alphabet and echoing property is supplied. This is a statement-level comparison, not an application of that theorem.

For the source convention at nonzero intercept, a separate caution is needed. Bugeaud–Laurent's printed page 2 displays an additional term $\beta^2\alpha\lfloor\theta+\rho\rfloor$ in the relation between its $\xi$ and $F$ series. From the printed definitions, put $T_n=\sum_{j=1}^{\lfloor n\theta+\rho\rfloor}\alpha^j$ and $T_0=0$. For $0<\theta<1$, $0\le\rho<1$, their difference is $T_n-T_{n-1}=s_n\alpha^{\lfloor n\theta+\rho\rfloor}$. In the domain $|\beta|<1$ and $|\beta|\,|\alpha|^\theta<1$, direct telescoping gives

$$\xi_{\mathbf s_{\theta,\rho}}(\beta,\alpha) =(1-\beta)F_{\theta,\rho}(\beta,\alpha).$$

This is a direct derivation, not a source-issued correction. The additional printed term is nonzero when $\theta+\rho\ge1$ and $\alpha,\beta\ne0$; it vanishes at the zero intercept used by the two-prime argument. This auxiliary display supplies no refutation of the numbered theorem.

Sketch and standing

The target is the transcendence, or at least the irrationality, of $S_P$ for fixed distinct primes $p,q,r$, with the original multiplicities.

Held links are the exact summand, carry partition, event and telescoping identities, with their elementary derivations above; the known [[research/smooth_lcm_three_prime/finite_separation|finite-separation obstruction]]; the complete author-recorded tier-0 arguments L189 and L190; and the exact finite formal relation searches. The arithmetic argument also holds at author-recorded standing: it identifies the genuine block numerators, proves quadratic growth of normalized tails, and rules out one direct fixed-base transfer. No unproved premise is used in these held links.

The separation route still lacks either an exact relation making $S_P$ a nonconstant algebraic function of one applicable transcendental value, or a relation to several values with enough joint arithmetic information to control the resulting expression. The excluded finite indicator partitions cannot supply it. A value theorem applying directly to the coupled sum is another possible bridge, but none of the inspected theorem statements covers it.

The arithmetic route lacks a non-cancelling family of small integer linear forms in the actual tails, or an applicable theorem for the variable integer denominator chain. Under a rationality assumption the normalized tails would be integral after one fixed multiplier, but their quadratic growth prevents a finite-state pigeonhole argument. A new estimate must use the actual smooth-count numerators, since other positive quadratic numerators on the same denominator chain give rational values.

These are precise gaps in the proposed arguments. They neither refute the irrationality target nor establish a no-go theorem for all representations. The wider finite-prime question remains outside the scope of the target here; where an obstruction is stated uniformly for larger sets, its proof is supplied separately in L189 or above.

Reopening the tested separations

A new identity can reopen the route if it falls outside the excluded finite termwise and same-coordinate halfspace classes and reaches a value theorem at the actual algebraic points and logarithmic slopes. Possible interfaces include an infinite partition with controlled summation, a representation with additional coordinates, or an identity holding only after specialization. None is furnished merely by naming the interface.

A theorem for the two-floor series $\sum_n\beta^n\alpha_1^{\lfloor n\theta_1\rfloor} \alpha_2^{\lfloor n\theta_2\rfloor}$ would also require an exact bridge to the full sum and a check of its actual slope conditions. In particular, a hypothesis about simultaneous rational independence must not be replaced by the pairwise irrationality that unique factorization supplies. For the arithmetic route, reopening the fixed-base transfer requires a different coefficient construction, or a theorem allowing the demonstrated coefficient heights and variable denominators.

Evidence

evidence/main.py runs, from the repository root, with uv run --no-sync python wiki/research/smooth_lcm_three_prime/evidence/main.py. It checks in exact arithmetic the running-lcm formula against a brute-force lcm at every smooth cutoff up to $10^{14}$ for five prime sets, finite forms of the telescoping identity and of the event formula, the examples at the cutoffs $30$ and $6$, explicit triangular face-rank witnesses at $(2,3,5)$ with $N\le6$, and the formal relation searches with their positive controls. The full run takes about two minutes and about 1 GB of memory; --quick takes about fifteen seconds and keeps only the smallest formal search. The checks cover finite ranges and families; the identities and obstructions are proved above and in the claims.