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
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
Gottfried Helms, Kassel