The Strong Lucas test coming in the next PR will already be complicated
enough. It'll be convenient, and less distracting, if we already have
functions for certain operations we'll need.
One thing we'll need to do is detect inputs that are perfect squares.
Fortunately, this is pretty easy to do robustly and quickly, with
Newton's method. We _don't_ want to use `std::sqrt`, because that takes
us into the floating point domain for no good reason, which could give
us wrong answers for larger integers.
The other thing we need is Jacobi symbols. These are a lot more
obscure, but thankfully, still resonably straightforward to compute.
The Wikipedia page (https://en.wikipedia.org/wiki/Jacobi_symbol) has a
good explanation, and in particular, good instructions for computing
values.
With these utilities in place, the Strong Lucas code should be easier to
review.
This can mark a number as either "probably prime", or "definitely
composite". The first parameter is the base, and the second is the
number to test.
Future PRs will build up the Strong Lucas test which complements this,
and then form the Baillie-PSW test by composing the two.
Helps #506.
The prime-testing techniques we will use (Miller-Rabin, Strong Lucas)
all make heavy usage of modular arithmetic. Therefore, we lay those
foundations here, adding utilities to perform the basic arithmetic
operations robustly.
Since these are internal-only helper functions, we don't bother checking
the preconditions, although we state them clearly in the contract
comment for each utility. After C++26, we could add contracts for
these.
Helps #509.
Chip Hogg
Mateusz Pusz