Let f be the formal power series over Z/2Z whose nth coefficient is independently chosen to be 1 with probability Ü(n^(

Definitions and mathematical statement

Let $\mathbb F_2=\mathbb Z/2\mathbb Z$, and let \[ f(X)=\sum_{n\geq0}a_nX^n\in\mathbb F_2[[X]],\qquad a_0=1. \] For $n\geq1$, suppose the coefficients $a_n$ are independent Bernoulli random variables with $\Pr(a_n=1)=p_n$ and $\Pr(a_n=0)=1-p_n$. The source describes $p_n$ using the expression "Ü$(n^{-2})$", whose leading symbol is not a defined probability or asymptotic operator in the supplied record. Thus the sequence $(p_n)$, including its admissible quantifiers, remains unspecified.

Since $a_0=1$, there is a unique formal reciprocal \[ \frac1{f(X)}=\sum_{n\geq0}b_nX^n\in\mathbb F_2[[X]], \qquad b_0=1,\qquad b_n=\sum_{k=1}^n a_k b_{n-k}\quad(n\geq1), \] where the last sum is in $\mathbb F_2$. Regard $b_n$ also as the integer $0$ or $1$. The source asks whether the density of ones in this reciprocal is $1/2$, namely whether \[ \lim_{N\to\infty}\frac1N\sum_{n=0}^{N-1}b_n=\frac12. \] If an almost-sure assertion is intended, the precise probabilistic target is that this limit exists and equals $1/2$ with probability one. The source does not explicitly specify almost-sure convergence, convergence in probability, or an assertion about expected densities, and the corrupted probability expression prevents fixing a complete probability law.

Short English statement

Choose independent binary coefficients with constant term one and study the formal reciprocal over $\mathbb F_2$. Must half its coefficients be ones in the limiting natural-density sense? The probability formula is corrupted and the mode of probabilistic assertion is unspecified.

Sources

• Cooper, Combinatorial Problems I Like, October 2020 snapshot, "Classical Number Theory," source-order bullet 72, https://people.math.sc.edu/cooper/combprob.html; accessed 2026-07-29, HTML extraction.
• The supplied research summary calls this a companion to source item 0071 associated with Riasanovsky; it supplies no further bibliographic citation or verified resolution.
• Local registry data/unsolvedmath/problems.json, permanent UnsolvedMath ID $3100072$, source code catalogue entry. Dataset attribution: Ulam AI and the UnsolvedMath Contributors.