论文标题
Farey Sequence和Graham的猜想
Farey sequence and Graham's conjectures
论文作者
论文摘要
令$ {f} _ {n} $为订单$ n $的farey序列。对于$ s \ subseteq {f} _n $，我们让$ \ Mathcal {q}（s）= \ left \ {x/y：x，x，y \ in s，x \ le y \，\，\，\，\ textrm {and} \，} \，\，\，\，\，\，\，y \ neq 0 \ neq 0 \ right \ right \ right \ right \ right \} $。我们表明，如果$ \ Mathcal {q}（s）\ subseteq f_n $，则$ | s | \ leq n+1 $。此外，我们在以下任何情况下证明：（1）$ \ Mathcal {q}（s）= f_n $; （2）$ \ MATHCAL {Q}（s）\ subseteq f_n $和$ | s | = n+1 $，我们必须有$ s = \ left \ weft \ {0,1，\ frac {1} {2} {2} {2}，\ ldots，\ ldots，\ ldot $ s = \ left \ {0,1，\ frac {1} {n}，\ ldots，\ frac {n-1} {n} {n} {n} \ right \} $除$ n = 4 $外第二种情况下，\ frac {2} {3} \} $。我们的结果基于Graham的GCD猜想，这些猜想已由Balasubramanian和Soundararajan证明。
Let ${F}_{n}$ be the Farey sequence of order $n$. For $S \subseteq {F}_n$ we let $\mathcal{Q}(S) = \left\{x/y:x,y\in S, x\le y \, \, \textrm{and} \, \, y\neq 0\right\}$. We show that if $\mathcal{Q}(S)\subseteq F_n$, then $|S|\leq n+1$. Moreover, we prove that in any of the following cases: (1) $\mathcal{Q}(S)=F_n$; (2) $\mathcal{Q}(S)\subseteq F_n$ and $|S|=n+1$, we must have $S = \left\{0,1,\frac{1}{2},\ldots,\frac{1}{n}\right\}$ or $S=\left\{0,1,\frac{1}{n},\ldots,\frac{n-1}{n}\right\}$ except for $n=4$, where we have an additional set $\{0, 1, \frac{1}{2}, \frac{1}{3}, \frac{2}{3}\}$ for the second case. Our results are based on Graham's GCD conjectures, which have been proved by Balasubramanian and Soundararajan.
