Iteration exercises: Lucas-Lehmer-test and Schröder-function - Printable Version +- Tetration Forum (https://math.eretrandre.org/tetrationforum) +-- Forum: Tetration and Related Topics (https://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=1) +--- Forum: Mathematical and General Discussion (https://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=3) +--- Thread: Iteration exercises: Lucas-Lehmer-test and Schröder-function (/showthread.php?tid=730) Iteration exercises: Lucas-Lehmer-test and Schröder-function - Gottfried - 04/04/2012 Since it is a bit quiet currently someone might enjoy an iteration exercise which I've applied to the Lucas-Lehmer-test for the primality of Mersenne-numbers. The Lucas-Lehmer-test *is* just an application of iteration of some simple function, but it is unusual to express this iteration using the concept of the Schröder-function and the Carleman-matrix. After the coefficients of the Schröder-functions have a simple pattern, that functions could be identified with the cosh and arccosh-functions; an identity which was also already known to Schröder himself and was also introduced and is mentioned in Chris Caldwell's nice Prime-pages. Here is my approach which led to a new "Lucal-Lehmer-Constant" L which allows to do the Lucas-Lehmer-test just by the test $\lceil L^{2^p} \rceil = 0 (mod M_p)$ and if the equality holds, then $M_p$ is prime.(Well, for p>7 we need so many digits of L that the test is not practical) Here is the link: lucasLehmerConstant Enjoy - Gottfried