SOLVED
This has been resolved in some other way than a proof or disproof.
Define a sequence by $a_1=1$ and\[a_{n+1}=\lfloor\sqrt{2}(a_n+1/2)\rfloor\]for $n\geq 1$. The difference $a_{2n+1}-2a_{2n-1}$ is the $n$th digit in the binary expansion of $\sqrt{2}$.
Find similar results for $\theta=\sqrt{m}$, and other algebraic numbers.
The result for $\sqrt{2}$ was obtained by Graham and Pollak
[GrPo70]. The problem statement is open-ended, but presumably Erdős and Graham would have been satisfied with the wide-ranging generalisations of Stoll (
[St05] and
[St06]).
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This page was last edited 28 September 2025.
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T. F. Bloom, Erdős Problem #482, https://www.erdosproblems.com/482, accessed 2026-01-14
