论文标题
广义斐波那契序列与坎宁安链的长度之间的关系
The relation between a generalized Fibonacci sequence and the length of Cunningham chains
论文作者
论文摘要
令$ p $为主要数字。链条$ \ {p，2p+1,4p+3，\ cdots，（p+1）2^{l（p）-1} -1} -1 \} $，如果所有元素均为prime数字，而$（p+1）2^{l（p）} - 1 $是组合的，则由$ p $生成的cunningham链。然后$ l（p）$称为坎宁安连锁店的长度。贝特曼（Bateman）和霍恩（Horn）在1962年猜想的是，$ l（p）\ geq k $的质量$ p \ leq n $的数量渐近地等于$ b_k n/（\ log log n）^k $，带有真正的$ b_k> 0 $，用于所有自然数量$ k $。这表明$ l（p）=ω（\ log p/\ log \ log p）$。但是，到目前为止尚无良好的估计。甚至没有证明$ \ limsup_ {p \ to \ infty} l（p）$是无限的。我们所知道的是，如果$ l（p）= 5 $如果$ p = 2 $和$ l（p）<p $ for odd $ p $ y fermat的小定理。令$α\ geq3 $为整数。在本文中，一个通用的fibonacci序列$ \ Mathcal {f}_α= \ {f_n \} _ {n = 0}^\ infty $定义为$ f_0 = 0，f_1 = 1，f_1 = 1，f_1 = 1，f_ {n+2} = n+2} $ {} _ {\ MATHCAL {f}_α}σ（n）= \ sum_ {d \ mid n，0 <d \ in \ Mathcal {f}_α} d $称为在$ \ MATHCAL {F}_α$上称为$ \ MATHCAL函数。然后，我们获得了$ {} _ {\ Mathcal {f}_α}σ$的迭代与坎宁安链的长度之间的有趣关系。对于两个Primes $ P $和$ Q $，事实$ P = 2Q+1 $或$ 2Q-1 $等于$ {} _ {\ Mathcal {f}_α}σ（{} _ {\ Mathcal {f}_α}σ（f_p））= {} _ {\ Mathcal {f}_α}_α}σ（F_Q）$α$。通过这种关系，我们在一定条件下获得$ l（p）\ ll \ log p $。通过数值测试，这种足够的条件似乎是合理的。此外，以质量数字编写的条件可以用自然数字所写的条件代替。这意味着$ L（p）$的上限估计的问题减少到自然数。
Let $p$ be a prime number. A chain $\{p,2p+1,4p+3,\cdots,(p+1)2^{l(p)-1}-1\}$ is called the Cunningham chain generated by $p$ if all elements are prime number and $(p+1)2^{l(p)}-1$ is composite. Then $l(p)$ is called the length of the Cunningham chain. It is conjectured by Bateman and Horn in 1962 that the number of prime $p\leq N$ such that $l(p)\geq k$ is asymptotically equal to $B_k N/(\log N)^k$ with a real $B_k>0$ for all natural number $k$. This suggests that $l(p)=Ω(\log p/\log\log p)$. However, so far no good estimation is known. It has not even been proven whether $\limsup_{p\to\infty} l(p)$ is infinite or not. All we know is that $l(p)=5$ if $p=2$ and $l(p)<p$ for odd $p$ by Fermat's little theorem. Let $α\geq3$ be an integer. In this article, a generalized Fibonacci sequence $\mathcal{F}_α=\{F_n\}_{n=0}^\infty$ is defined as $F_0=0,F_1=1, F_{n+2}=αF_{n+1}+F_n (n\geq0)$, and ${}_{\mathcal{F}_α}σ(n)=\sum_{d\mid n, 0<d\in\mathcal{F}_α}d$ is called a divisor function on $\mathcal{F}_α$. Then we obtain an interesting relation between the iteration of ${}_{\mathcal{F}_α}σ$ and the length of Cunningham chains. For two primes $p$ and $q$, the fact $p=2q+1$ or $2q-1$ is equivalent to ${}_{\mathcal{F}_α}σ({}_{\mathcal{F}_α}σ(F_p))={}_{\mathcal{F}_α}σ(F_q)$ for some $α$. By this relation, we get $l(p)\ll\log p$ under a certain condition. It seems that this sufficient condition is plausible by numerical test. Furthermore, the condition, written in terms of prime numbers, can be replaced by the condition written in terms of natural numbers. This implies that the problem of upper estimation of $l(p)$ is reduced to that on natural numbers.
