论文标题
在持续的分数部分的正方形根源
On continued fraction partial quotients of square roots of primes
论文作者
论文摘要
我们表明，对于每个正整数$ a $，只有有限的许多质数$ p $，因此$ a $在$ \ sqrt {p} $或$ \ sqrt {2p} $的持续分数中出现奇数次。我们还证明，如果$ p $是质量数字，而$ d = p $或$ 2p $是这样，以至于$ \ sqrt {d} $的持续分数扩展期间的长度可将$ 4 $整理，那么$ 1 $在$ \ sqrt {d} $的持续分数中作为部分商品出现。此外，我们在$ \ sqrt {d} $的持续分数扩展的周期长度上给出了上限，其中$ d $是一个积极的非方面，并将某些与正方形的正方根持续分数相关的多项式系数分配给某些多项式系数。这些结果回答了Miska和Ulas最近提出的几个问题。
We show that for each positive integer $a$ there exist only finitely many prime numbers $p$ such that $a$ appears an odd number of times in the period of continued fraction of $\sqrt{p}$ or $\sqrt{2p}$. We also prove that if $p$ is a prime number and $D=p$ or $2p$ is such that the length of the period of continued fraction expansion of $\sqrt{D}$ is divisible by $4$, then $1$ appears as a partial quotient in the continued fraction of $\sqrt{D}$. Furthermore, we give an upper bound for the period length of continued fraction expansion of $\sqrt{D}$, where $D$ is a positive non-square, and factorize some family of polynomials with integral coefficients connected with continued fractions of square roots of positive integers. These results answer several questions recently posed by Miska and Ulas.
