Reciprocal Mersenne Subseries

Will Cook

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Reciprocal Mersenne Subseries

Erdős #257 By Will Cook 30 September 2026 Authorship, AI use and citation

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Summary. A weighted summability criterion proves irrationality of reciprocal Mersenne subseries: if h is the part of a supported on a finite nonempty prime set P, every infinite A with ∑ $h (a) / (a (b^{h (a)} - 1))$ finite has irrational ∑ $(b^{a} - 1)^{- 1}$ . The base-two condition gives the conclusion at every integer base for every infinite subset, and permits an example with divergent reciprocal sum, while excluding full support and all odd exponents. The proof averages over a finite part of the support and over dyadic observation lengths. Erdős's weaker reciprocal-summable theorem is proved in full, and a common window extends both arguments to unions of weighted and positive-cover supports. Finite-denominator and achievement-set calculations formulate tests for the targets $1 / 2$ and $1 / 21$ ; those tests do not decide either target or the universal question. Prime support at base 2 and squarefree support at power-of-two bases are attributed as existing work.

This paper covers a claim-bounded problem-specific reasoning surface for Erdős #257: selected checked premises, finite computations, closed routes, surviving obligations, and their exact evidence bands.

It does not establish the validity of claims tagged Lean, which belongs to the cited kernel-checked source, or a solution to Erdős #257, whose universal and half-value questions remain open.

Start here, by the editorial route

In this paper

Known support theorems

Full-support irrationality is Erdős’s theorem [1]. The pairwise-coprime case is Erdős’s theorem of 1968, where he also stated that the coprimality condition can be removed [2]. Luca and Tachiya proved irrationality for every purely periodic integer weight that is not identically zero [3]; the purely periodic, eventually periodic, residue-class, odd-support and periodic-weight rows below follow from their theorem, and odd support is their Example 2 [3]. The lcm-gap supports, including factorials and powers of two, have summable reciprocals, as the calculation following the table shows. They are consequently included in the reciprocal-summable theorem stated by Erdős. The final column locates the corresponding declarations.

The two kinds of argument in the table use different data: divisibility of blocks of $c_{A} (n)$ , or rational approximations obtained from a growing lcm gap. The hypotheses are stated separately for each row. The notation ${sc}_{A} (n)$ in later source excerpts means the same function as $c_{A} (n)$ . All bases in the table are integers $b \geq 2$ . For the lcm-gap row write $L_{k} = lcm (a_{0}, \dots, a_{k - 1})$ , with $L_{0} = 1$ .

Support class	Exact hypotheses	Formal proof
Full support $A = N_{> 0}$	$b \geq 2$	`irrational_erdosSum_full_support`
Multiples $A = d N_{> 0}$	$b \geq 2$ , $d \geq 1$	`irrational_erdosSupportSeries_multiples`
Purely periodic	$m \geq 1$ , $m$ -periodic, contains a positive element	`irrational_erdosSupportSeries_periodic`
Eventually periodic	$m \geq 1$ , $m$ -periodic from $N_{0}$ , $A$ infinite	`irrational_erdosSupportSeries_eventuallyPeriodic`
Residue class	any $m \geq 1$ , any $c$	`irrational_erdosSupportSeries_residueClass`
Odd support (density $1 / 2$ )	$b \geq 2$	`irrational_erdosSupportSeries_odd`
Growing lcm gap	$a_{k}$ strictly increasing and positive; $a_{k} - L_{k} \to \infty$	`irrational_erdosSum_of_lcm_gap`
Factorials; powers of two	instances of the lcm-gap theorem	`irrational_erdosSum_factorial_support`, `irrational_erdosSum_two_pow_support`
Pairwise coprime (Erdős 1968)	$A$ infinite, pairwise coprime, and $\sum_{a \in A} 1 / a < \infty$	`irrational_erdosSupportSeries_pairwise_coprime`
Signed periodic weights	periodic integer weights; their divisor-transformed coefficients are nonnegative and frequently nonzero	`irrational_intWeightedErdosSeries_periodic_of_coeff_nonneg_of_frequently_ne_zero`

The lcm-gap hypothesis does imply reciprocal summability. Eventually $a_{k + 2} > lcm (a_{k}, a_{k + 1})$ . If $a_{k + 1} \geq 2 a_{k}$ , monotonicity gives $a_{k + 2} > 2 a_{k}$ . Otherwise $a_{k} ∤ a_{k + 1}$ , so $lcm (a_{k}, a_{k + 1}) \geq 2 a_{k + 1} > 2 a_{k}$ . Thus each of the even- and odd-indexed subsequences eventually grows geometrically, proving $\sum_{k} 1 / a_{k} < \infty$ . This explains the inclusion claimed above, rather than using “lacunary” as a substitute for a growth hypothesis. For eventually periodic weights, the periodic tail must not be identically zero: an eventually zero weight sequence gives only a finite rational sum.

For $b = 2$ , the first row is the irrationality of the Erdős–Borwein constant $E = \sum_{n \geq 1} (2^{n} - 1)^{- 1}$ . The all-base scope belongs to Erdős’s 1948 theorem; this record contributes its formalisation. The other rows retain their displayed restrictions and leave arbitrary infinite support open. A block-certificate argument may additionally require independence or a tail estimate for its particular support. Barrier 2.7 discusses those construction requirements; the averaging proof above does not assume pairwise coprimality or independence of divisor events.

A separate conditional result allows common factors confined to divisors of one fixed integer $Q$ , while the remaining factors must be pairwise coprime and have summable reciprocals (irrational_erdosSupportSeries_of_orthogonalPetalBouquet). It also assumes that for every $K \geq 1$ there is an $N \geq 0$ such that

2^{K} ∣ \sum_{r = 1}^{K} c_{A} (N + r) 2^{K - r}, \sum_{r \geq 1} c_{A} (N + K + r) 2^{- r} \leq 16.

Thus relaxing coprimality does not remove the simultaneous divisibility and tail-bound requirement. This additional hypothesis is not proved here, and the source file does not establish a dichotomy for arbitrary supports.

The record also gives a logarithmic bound on runs of zero divisor counts under rationality: for each $ε > 0$ , a suitable constant $B$ bounds the length by $ε \log_{2} N + B$ beyond the stated threshold (supportCoeffZeroWindow_length_le_eps_logb). For these supports there is already a simpler unconditional bound. If $a_{0} = min A$ , every $a_{0}$ consecutive positive integers contain a multiple of $a_{0}$ , where $c_{A}$ is positive. Hence every zero run has length at most $a_{0} - 1$ , whether or not $X_{A} (2)$ is rational. In particular the logarithmic bound alone does not distinguish rational-valued supports. The separate unbounded-tail result shifted_state_unbounded_of_infinite_support also requires interpretation: an unbounded sequence can still satisfy $u (N) = o (2^{N})$ .

Remark 1.7 (What is and is not covered in the universal direction). The Lean sources contain no theorem for $A =$ the primes, but this is a formalisation boundary, not a mathematical open case: Tao and Teräväinen prove the base- $2$ prime-support value irrational by first identifying it with $\sum_{n \geq 1} ω (n) 2^{- n}$ . Duverney and Tachiya likewise settle squarefree support at every power-of-two base, and the perfect $i$ th powers at every integer base (Corollary 1.2 and Example 1.1, p. 4 of the author preprint). Neither result is formalised here. The statement of Problem #257 has been formalised before, as a conjecture with an unfilled proof, in the Formal Conjectures collection; that file also proves the Lambert identity for the full-support value and states its irrationality, the 1948 theorem of Erdős, with proof sorry. It is statement-level prior art, and no theorem recorded here is derived from it. This record has no general theorem for an arbitrary positive-density nonperiodic or divisor-dense support, and the universal statement remains open.

Periodic weights: the two hypothesis sets.

The periodic sequence theorem belongs to Luca and Tachiya [14]; their later author-written account states it as Theorem A [3]. In the displayed Lean row above, nonnegativity and frequent nonvanishing are assumptions on $c_{w} (n) = \sum_{d ∣ n} w (d)$ , not on $w (n)$ itself. The formal row must therefore be read as a separately specified instance, not as an exact transcription of every clause of the published theorem.

A separate denominator-growth criterion.

Erdős’s growth theorem [21], as restated and recovered by Barreto, Kang, Kim, Kovač and Zhang [22], gives irrationality of $\sum_{n} 1 / a_{n}$ for an increasing integer sequence with $a_{n} \geq n^{1 + τ}$ for some $τ > 0$ and $\underset{n}{lim sup} a_{n}^{1 / 2^{n}} = \infty$ . For $a_{n} = b^{c_{n}} - 1$ , where $c_{n}$ are increasing positive exponents and $b \geq 2$ is an integer, the growth hypothesis becomes

\underset{n \to \infty}{lim sup} \frac{c_{n}}{2^{n}} = \infty .

Indeed $c_{n} \geq n$ , so the polynomial lower bound holds eventually, and $\log_{b} (b^{c_{n}} - 1) = c_{n} + O (1)$ . An eventual lower bound suffices: deleting finitely many terms subtracts a rational number and changes $2^{n}$ only by a fixed factor after reindexing.

The two hypotheses can be separated explicitly. The squares $c_{n} = n^{2}$ have convergent reciprocal sum but $c_{n} / 2^{n} \to 0$ . In the other direction, set $n_{1} = 1$ and $n_{k + 1} = n_{k} + 4^{n_{k}}$ , and let block $k$ consist of all integers from $4^{n_{k}}$ to $2 \cdot 4^{n_{k}} - 1$ . The blocks are successive disjoint increasing blocks, and their union has increasing enumeration satisfying $c_{n_{k}} = 4^{n_{k}}$ . Thus $c_{n_{k}} / 2^{n_{k}} = 2^{n_{k}} \to \infty$ , whereas each block has reciprocal sum at least $1 / 2$ . Its reciprocal sum therefore diverges. These examples compare denominator growth with reciprocal summability only; they leave the other support criteria out of the comparison.

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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