Running least common multiples: distinct values and multiplicities

Will Cook

Long paper

Running least common multiples: distinct values and multiplicities

Erdős #269 By Will Cook July 2026; revised 5 October 2026 Authorship, AI use and citation

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Summary. For the prime set {2,3,5}, a five-map argument proves irrationality of the sum over distinct running LCMs. Two-prime sums reduce to a transcendental Hecke–Mahler value. For the repeated three-prime sum, an equivalent residue criterion leaves an arithmetic condition open. Complete ordinary arguments for general distinct-running-LCM sums and prime-power subseries are retained as remarks awaiting formalisation.

This paper covers distinct-height irrationality, all-rank kernel structure and the repeated-sum tail criteria, preserving the multiplicity distinction and the remaining open residue-window obligation.

It does not establish the validity of claims tagged Lean, which belongs to the cited kernel-checked source, or a solution to Erdős #269, which remains open.

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In this paper

Source inventory

The table gives declaration locations at their recorded revisions. Section 12.5 explains the assumptions attached to each kind of association. Before using the table, two conventions require attention: integer division at the initial index and removal of the first block. The Lean version of the normalising factor $h_{a}$ is a natural number computed by integer division: its value at $0$ is $0$ , whereas $h_{0} = 1 / 2$ . It agrees with $h_{a}$ for $a \geq 1$ , as the identity for half the running LCM assumes. The Lean version of the tail $X_{a}$ divides after casting to $R$ , so agrees with $X_{a} = h_{a} T_{a}$ at every index.

The denominator-clearing lemmas and the lemmas on division of an integer recurrence assume $T_{1}$ rational. The first block consists of $1$ alone, so $T_{1} = S - 1$ ; thus $S = N / D$ supplies

T_{1} = \frac{N - D}{D}, gcd (N - D, D) = gcd (N, D) .

Lean proves this translation, which preserves the denominator, its smooth factor and the clearing onset.

informal statement	linked Lean source
integer represented by an exponent triple	smooth3 val
pure-power running LCM	three prime height
lattice kernel	three prime kernel q
exponent triples below the cutoff	smooth prefix exponents
running least common multiple	smooth prefix lcm
prefix value divides the running LCM	smooth3 val dvd three prime height of mem
divisibility into the running LCM	smooth prefix lcm dvd three prime height
first pure-power membership	pure first mem smooth prefix exponents
second pure-power membership	pure second mem smooth prefix exponents
third pure-power membership	pure third mem smooth prefix exponents
running-lcm identity	smooth prefix lcm eq three prime height
cell relation	same three prime log cell
cell constancy of the running LCM	three prime height eq of same log cell
cell constancy of the running value	smooth prefix lcm eq of same log cell
cell constancy of the kernel	three prime kernel q eq of same log cell
positive power sets	positive prime powers
counting positive prime powers	positive prime powers card
exclusion of the origin	one not mem positive prime powers
disjoint positive prime powers	positive prime powers disjoint
union of positive prime powers	three prime positive jump set
positive jump count	three prime positive jump set card
jump set with the origin	three prime jump set with origin
jump count with the origin	three prime jump set with origin card
first running LCM step	three prime height first log step
second running LCM step	three prime height second log step
third running LCM step	three prime height third log step
first coordinate step	smooth prefix lcm first log step
second coordinate step	smooth prefix lcm second log step
third coordinate step	smooth prefix lcm third log step
exponent box	smooth exponent box
point running LCM	smooth point height
running LCM fibre	smooth height fiber
fibre sum	smooth height fiber kernel sum
grouping equal denominators in a finite sum	finite smooth kernel sum grouped by height
upper bound by the cube of the cutoff	three prime height le cube
origin value	kernel 235 origin
value at two	kernel 235 two
value at three	kernel 235 three
value at six	kernel 235 six
example excluding a product of two factors	kernel 235 not rank one
variable-base tail step	tail state step
expanded tail step	tail state step eq
smooth exponent block	smooth exponent shell
short-interval uniqueness	exponent unique in short interval
first-coordinate projection	smooth exponent shell card le drop first
third-coordinate projection	smooth exponent shell card le drop third
sorted quadratic estimate	sorted pair quadratic
quadratic block bound	smooth exponent shell card quadratic
dyadic internal power	dyadic internal power
internal-power uniqueness	dyadic internal power exponent unique
dyadic block base	dyadic block base235
exact dyadic radix alphabet	dyadic block base235 cases
bounded-radix consequence	dyadic block base235 mem interval
a nonzero $2 \times 2$ determinant	kernel 235 minor eq neg one fifteen
no integer rotation orbit	no integer orbit logb of prime
running LCM factorisation	three prime height factorisation
kernel factorisation	three prime kernel q factorisation
two-prime outer product	two prime kernel q eq outer product
two-prime vanishing minors	two prime kernel q minor two eq zero
uniform minors, general form	exists uniform nonsingular three prime kernel minor
uniform minors for primes	exists uniform nonsingular three prime kernel minor of prime
no finite separation	not finite separable three prime kernel
second-order minor	kernel 235 minor2 eq neg one fifteen
third-order minor	kernel 235 minor3 eq one over 81000
misleading proportional row	kernel 235 row three eq smul row zero
failure of that proportionality	kernel 235 row three ne smul row zero at four
integer numerator for one dyadic block	dyadic ordered block digit235
recurrence for normalised tails	dyadic normalized tail state r235 succ
block summability	summable dyadic shell mass r235
infinite tail recurrence	dyadic normalized shell tsum tail r235 succ
integer tails or separation from the integers	dyadic shell tsum tail integer or cofinal far
boundary divisibility	smooth height mul prime dvd boundary height
half the running LCM for $a \geq 1$	two mul height normalizer235
segment clearing identity	height normalizer235 mul window mass eq int
clearing a rational $T_{1}$ at $a \geq 1$	qsmul normalized tail state eq int of value eq rat
congruence between normalised tails	exists normalized tail state collision of value eq rat
least positive residue	least positive residue
positive representative range	least positive residue pos le
representative congruence	least positive residue mod eq
residue inequality	residue escapes window
excluding a bounded positive natural number	no bounded positive state of residue escape
contrapositive form	residue le bound of bounded positive state
excluding a bounded positive integer	no bounded positive int state of least positive residue
identifying the least positive representative	least positive residue eq nat abs of pos le mod eq
product of the bases	window base
accumulated numerator in a segment	window forcing
affine iterated recurrence	affine recurrence window
scaling the accumulated numerator	window forcing const mul
iterating an integer recurrence	integral carry window
endpoint residue identification	least positive residue window forcing eq carry
divisibility by the smooth part of the denominator	smooth3 val dvd three prime height of le
common-factor cancellation	integral carry cancel common factor
bound after dividing out the common factor	reduced carry pos le of common factor
iterated recurrence after division	reduced integral carry window
cofinal segment hypothesis	cofinal local window escape
the residue inequality excludes a positive integer solution of the reduced recurrence	no positive reduced carry of cofinal local window escape
the same contradiction with a common factor	no positive absorbed carry of cofinal local window escape
real iterated recurrence	true normalized state window
irrationality implies the residue inequality, general bound	cofinal local window escape of irrational of beaten
irrationality implies the residue inequality, quadratic family	cofinal local window escape of irrational of quadratic
irrationality implies the residue inequality, actual bound	cofinal local window escape of irrational
residue criterion equivalent to an irrational tail	actual cofinal local window escape iff
residue criterion equivalent to an irrational sum	actual cofinal local window escape iff irrational value
rational $T_{1}$ gives eventual integer carries	exists reduced carry of value eq rat
residue condition for the actual series	actual cofinal local window escape
irrationality from the residue condition	irrational value of cofinal local window escape
bounded-radix alternative	bounded radix zero or cofinal far
rational value from an integral scaled tail	rational of scaled tail integer
zero block sums as differences of potentials	channel block null iff channel potential
a nonzero error exceeds the bound at one index	no carry lift of error bound below two pow
a nonzero error cannot stay uniformly bounded	no bounded carry lift of block null two anchors
first-block sum	binary lift two anchors force first block sum one
four consecutive values cannot meet all three conditions	no unit accurate lift with two anchors and first two block null
carry as a residue plus a potential difference	carry eq residue digit add coboundary
carry interval	carry residue mem interval
digit interval	residue digit mem interval
map from formal expressions to functions	cartier realisation
finite-dimensional span of the resulting functions	finite realised span of factorisation
interval containing a Cantor-series tail	state mem ioo
lower bound for a difference of tails	state gap lower bound
small-form criterion	irrational of small nonzero forms
irrationality from clearing and small nonzero gaps	irrational of clearing and small gaps
indices giving the threshold staircase	staircase indices
finite threshold-column rank, including the extreme-column defect	finite cut rank
uniform admissible modular minors, with indices chosen before modulus and layer	uniform modular minors

About this paper

Authorship and AI use. Will Cook built and directed the research infrastructure and maintains the public release. He reviewed claims when he could. AI agents did most of the research and drafting. Cook did not independently verify every claim.

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