# What are some basic properties of the quotient $s_n\over M_p$ where $M_p$ is a Mersenne prime?

I've been exploring the Lucas-Lehmer test for a while now. I already know the square of one Mersenne $(2^n-1)^2$ is $(2^{n-1}-1)(2^{n+1}-1)+2^{n-1}$. Today I'm looking to use the properties of the other divisor of $s_{n-2}$, such that I can start the Lucas-Lehmer test from last Mersenne primes test. I know a bit about the other divisors already, like because the Lucas-Lehmer test is done with numbers that are 2 mod 4 and 3 mod 4 that the other divisor when it exists is 2 mod 4, and depending on n ( when it's prime) either 2 or 3 mod 7, and such (I haven't built the full list of residue classes for other known Mersenne primes yet). My attempt is for things like: $$s_{n-2}=a\cdot M_n$$ to be able to change to the next steps mod other Mersenne numbers. Any help ( even if I can't understand it quite yet), will be appreciated.

• as an example $14=2\cdot 7; 194=2^2\cdot 7^2-2 = 4 \cdot 4-2 =14 \pmod {15}$ – user451844 Aug 7 '17 at 1:26
• @RoddyMacPhee I'm not sure that mersenne-math is a good choice for a tag (it's vague, including "math" in a tag on MSE isn't really necessary, etc.). A Mersenne number tag might be useful, but I'd suggest putting it up here for discussion first - there are many questions involving Mersenne numbers without a dedicated tag. – user296602 Aug 7 '17 at 1:32
• they would probably be put under repunit tags, and if math in a tag isn't necessary there's at least 4+ tags you should alter then. – user451844 Aug 7 '17 at 1:35
• The LLT is based on the congruence $s_{p-2} \bmod 2^p -1$. Sure you can compute $s_{p-2}$ itself, but in practice if $p > 20$ you'll compute only $s_{p-2} \bmod 2^p -1$. – reuns Aug 7 '17 at 1:40
• @RoddyMacPhee Sure, there are other tags with issues; but in general, it's good practice to discuss proposed new tags in the meta thread I linked. Otherwise, there's a serious tendency to pick up tags that fragment rather than organize. A tag called "mersenne-primes" might be a lot better, but given that the community hasn't created it yet (with almost 600 related questions) indicates that it should probably be discussed first. – user296602 Aug 7 '17 at 1:59