Genius implements modular arithmetic.
To use it you just add "mod <integer>" after
the expression. Example:
2^(5!) * 3^(6!) mod 5
It could be possible to do modular arithmetic by computing with integers and then modding in the end with
% operator, which simply gives the remainder, but
that may be time consuming if not impossible when working with larger numbers.
10^(10^10) % 6 will simply not work (the exponent
will be too large), while
10^(10^10) mod 6 is instantaneous. The first expression first tries to compute the integer
10^(10^10) and then find remainder after division by 6, while the second expression evaluates
everything modulo 6 to begin with.
You can calculate the inverses of numbers mod some integer by just using rational numbers (of course the inverse has to exist). Examples:
10^-1 mod 101 1/10 mod 101
You can also do modular evaluation with matrices including taking inverses, powers and dividing. Example:
A = [1,2;3,4] B = A^-1 mod 5 A*B mod 5
This should yield the identity matrix as B will be the inverse of A mod 5.
genius> sqrt(4) mod 7 = [2, 5] genius> 2*2 mod 7 = 4
sqrt will actually return all the possible square
Do not chain mod operators, simply place it at the end of the computation, all computations in the expression on the left
will be carried out in mod arithmetic. If you place a mod inside
a mod, you will get unexpected results. If you simply want to
mod a single number and control exactly when remainders are
taken, best to use the
% operator. When you
need to chain several expressions in modular arithmetic with
different divisors, it may be best to just split up the expression into several and use
temporary variables to avoid a mod inside a mod.