论文标题
吉尔布雷斯猜想的随机类似物
A random analogue of Gilbreath's conjecture
论文作者
论文摘要
吉尔布雷斯（Gilbreath）的一个众所周知的猜想，并从1800年代独立耕种，指出，如果$ a_ {0，n} = p_n $表示$ n^{\ text {text {th}} $ prime number和$ a_ {i，n} $ a_ {i，1} = 1 $ for All $ i \ ge 1 $。有人反复假设，拥有$ a_ {i，1} = 1 $ for $ i $的属性应适用于任何初始$（a_ {0，n}）_ {n \ ge 1} $，规定gaps $ a__ {0，n+1} -a_} -a__ {0，n} $不够大，并且不够大。我们证明了这一假设的（精确形式）。
A well-known conjecture of Gilbreath, and independently Proth from the 1800s, states that if $a_{0,n} = p_n$ denotes the $n^{\text{th}}$ prime number and $a_{i,n} = |a_{i-1,n}-a_{i-1,n+1}|$ for $i, n \ge 1$, then $a_{i,1} = 1$ for all $i \ge 1$. It has been postulated repeatedly that the property of having $a_{i,1} = 1$ for $i$ large enough should hold for any choice of initial $(a_{0,n})_{n \ge 1}$ provided that the gaps $a_{0,n+1}-a_{0,n}$ are not too large and are sufficiently random. We prove (a precise form of) this postulate.
