论文标题
Chebyshev多项式和高阶Lucas Lehmer算法
Chebyshev polynomials and higher order Lucas Lehmer algorithm
论文作者
论文摘要
我们将Lucas Lehmer迭代的必要部分扩展到将Mersenne Prime测试到所有碱基，并均匀地用于广义的Mersenne和Wagstaff数字（后来对应于负基碱）。二次迭代$ x \ rightArrow x^2-2 $的作用由Chebyshev多项式$ T_N（X）$扩展，并具有隐含的迭代算法，因为构图身份$ t_n（t_m（x））= t_m（x）= t_ {nm}（nm}（x）$。这是由Chebyshev多项式原始测试基本上基于Lucas对$（ω_a，\overlineΩ_a）$，$ω_a= a+\ sqrt {a^2-1} $，其中$ a \ neq 0 \ pm 1 $。有趣的是，算术全部均在Chebyshev多项式$ T_N（X）$中编码。
We extend the necessity part of Lucas Lehmer iteration for testing Mersenne prime to all base and uniformly for both generalized Mersenne and Wagstaff numbers(the later correspond to negative base). The role of the quadratic iteration $x \rightarrow x^2-2$ is extended by Chebyshev polynomial $T_n(x)$ with an implied iteration algorithm because of the compositional identity $T_n(T_m(x))=T_{nm}(x)$. This results from a Chebyshev polynomial primality test based essentially on the Lucas pair $(ω_a,\overlineω_a)$, $ω_a=a+\sqrt{a^2-1}$, where $a \neq 0 \pm 1$. It seems interesting that the arithmetic are all coded in the Chebyshev polynomials $T_n(x)$.
