Problem Statement
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.
Find similar results for $\theta=\sqrt{m}$, and other algebraic numbers.
Categories:
Number Theory
Progress
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]).Source: erdosproblems.com/482 | Last verified: January 15, 2026