---

Claim. The request of Problem 482, results like the Graham--Pollak identity for $\sqrt m$ and other algebraic numbers, is answered for every positive real $w$. Theorem 1.2 of the 2005 paper gives, for every $w>0$ and every integer $j\ge1$, two families of two-step recurrences $u_1=1$, $u_{n+1}=\lfloor a(u_n+\varepsilon_n)\rfloor$, whose multiplier alternates between $a$ and $b=2/a$ with the parity of $n$ and whose shifts are $1/2$ (one family allows any $\varepsilon\in[1/3,2/3)$ on even steps), such that the differences $u_{2n+1}-2u_{2n-1}$ are the binary digits of $w$; the Graham--Pollak recurrence is the case $j=1$, $\varepsilon=1/2$, $w=\sqrt2$ of the family allowing any $\varepsilon$ (Case I). Theorem 1.3 does the same in every integer base $g\ge2$, with $u_{2n+1}-gu_{2n-1}$ the base-$g$ digits of $w$. Theorem 3.1 of the 2006 paper extends the construction to integer starting values $m$ and integer parameters $(l,k)$, with $(m,l)$ in six explicit cones, which require $m\ge1$ or $m\le-2$, and $(g-1)\mid(k-1)l$; Theorems 3.3 and 3.4 give two further binary families; and Corollary 3.5 identifies, for every integer $m\notin\{-1,0\}$, which number's binary digits the original recurrence with $u_1=m$ produces. The statements are quoted on the result pages theorem_1_2, theorem_1_3, theorem_3_1, theorem_3_3 and corollary_3_5; the digests are on the source cards stoll_2005_families_nonlinear_recurrences_related_digits and stoll_2006_problem_erdos_graham_concerning_digits. Algebraicity plays no role: the theorems hold for all of $\mathbb R^+$, so every $\sqrt m$ and every positive algebraic number is covered. They construct families and do not classify every recurrence with the digit property, which the request never asked for. The inductive proofs were read for structure only and are not checked here.

Claim value. The request names no proposition, so its resolution is neither a proof nor a disproof; the claim is recorded as answered, the outcome of a find question, which the site labels SOLVED.

Acceptance. Refereed: Th. Stoll, On families of nonlinear recurrences related to digits, J. Integer Seq. 8 (2005), Article 05.3.2, 8 pp. (received 1 April 2005, published 24 May 2005), and Thomas Stoll, On a problem of Erdős and Graham concerning digits, Acta Arith. 125 (2006), no. 1, 89--100 (received 16 March 2006). Reviewed: the site's curator, Thomas F. Bloom, marks Problem 482 SOLVED and names Stoll's generalizations in the problem's commentary as the resolution Erdős and Graham would presumably have accepted (page last edited 28 September 2025); the curator neither wrote nor submitted the result. Third-party Lean formalizations of these theorems are linked above: Trevor Morris's gallery, formalized with Claude Code, and Boris Alexeev's repository, whose formal authors are Codex and GPT-5.6 Sol. They were not built here, so no formalized evidence is listed. The page is dated by the publication date of the 2005 paper, the result's first posting.