#OK to post homework
#Joseph Koutsoutis, 04-27-2025, Assignment 24

read `C24.txt`:

#1

#I've written procedures for generating random matrices of the following types:

#symmetric, antisymmetric, tridiagonal, uppertriangular
#Toeplitz, Hankel, circulant, tridiagonal Toeplitz

#Except for circulant matrices, 2x2 and 3x3 matrices of the other types
#required 4 and 6 matrices respectively, matching the Amitsur–Levitzki theorem.

#It is known that circulant matrices commute. I started to experiment with
#matrices similar to circulant matrices with a different cyclic shift.
#It appears that anticirculant matrices (where the shift is -1 instead of +1)
#require exactly 3 matrices for n>=3. I found that this held for 3 <= n <= 50
#by testing symbolic anticirculant matrices.
#I also just started to experiment with cyclic shift +2.
#For n=3,4,5,6,7,8 it appears that m=3,4,5,4,3,5 matrices are sufficient.

#I'll probably try to experiment more with these types of matrices and will also
#return to some of the other types to see the effects of powers, zeroing out entries,
#and requiring elements to be the same on certain diagonals.

#I also want to see if I can at least prove the anticirculant case which I haven't tried to do yet.
#I have no idea if this is trivial but I couldn't find a proof online during a very brief search.

#2

CompareFibMod := proc(INI,M) local m:
  add(x[evalb(nops(FibMod(INI, m)[2]) > EstimateGFperiod(m^2,x,10000)[2][1])], m=2..M):
end:

#CompareFibMod([0,1], 100) outputted 12*x[false] + 87*x[true]