Reading Eight Erdős Problems Together

From individual problems to reusable questions

Seven of the eight programmes ask whether a series is irrational: $\sum (n!-1{)}^{-1}$ in #68, reciprocal sums of near-Sylvester sequences in #243, $\sum \phi (n)2^{-n}$ in #249, $\sum p_n{2}^{-n}$ in #251, $\sum _{n\in S}(2^n-1{)}^{-1}$ in #257, reciprocal running least common multiples in #269, and $\sum (t^n-1{)}^{-1}$ at rational $t$ in #1049 [6][7]. The eighth, #1041, concerns polynomial lemniscates. It supplies no premise for the series arguments below.

What can the freedom to choose digits preserve, and when does arithmetic remove that freedom? Sparse corrections in the #251 paper motivate the capacity criterion of Section 3. Its proof makes one interval of possible continuations work for every cumulative residue. The factorial examples show that the arithmetic restrictions change a sharp support-gap threshold. Factorial denominators $n!$ in this construction are not the denominators $n!-1$ in #68.

Section 2 asks whether a single factorial digit sequence can prescribe several derivatives independently. Theorems 2.1 and 2.2 give the sharp threshold and the dimension of the attainable vectors. The proof uses the scalar capacity theorem first, then a carry that preserves lower derivatives; Sections 4 and 5 give the argument and the rationality obstruction.

The common Lambert series in #257 and #1049 gives a second comparison in Section 6: a base below two permits rational subsums, while on every support with separated divisibility cuts the sum is transcendental at every real algebraic base greater than one, even with bounded positive integer weights. Section 6.3 gives the proof and a non-chain host. Neither interval filling nor a measure estimate decides whether one specified rational is a subsum at base two. Sections 6.4 and 9 explain that obstruction, including the exact computation that disproved a proposed stopping rule. Section 8 retains the other method limits without claiming that they have one common cause.

Section 7 follows a different transfer. A difference of two dyadic tail states is itself an integer-digit dyadic orbit. The resulting irrationality criterion leads to a question about which shift lengths need testing; the answer depends on divisibility rather than the size or density of the chosen family.