# OK to post homework
# Aurora Hiveley, 4/15/25, Assignment 22

Help := proc(): print(``): end:

# Run ConfirmR(2^n,19) for n from 10 to 20. Are all the denominators always 19?

# [seq(ConfirmR(2^n,19),n=10..20)]:
# [[1/19, 3/28, 4/25, 4/19, 5/19, 6/19, 7/19, 8/19, 9/19, 10/19, 11/19, 12/19, 13/19, 14/19, 15/19, 16/19, 17/19, 18/19], [1/19, 2/19, 3/19, 4/19, 5/19, 6/19, 7/19, 8/19, 9/19, 10/19, 11/19, 12/19, 13/19, 14/19, 15/19, 16/19, 17/19, 18/19], [1/19, 2/19, 3/19, 4/19, 5/19, 6/19, 7/19, 8/19, 9/19, 10/19, 11/19, 12/19, 13/19, 14/19, 15/19, 16/19, 17/19, 18/19], [1/19, 2/19, 3/19, 4/19, 5/19, 6/19, 7/19, 8/19, 9/19, 10/19, 11/19, 12/19, 13/19, 14/19, 15/19, 16/19, 17/19, 18/19], [1/19, 2/19, 3/19, 4/19, 5/19, 6/19, 7/19, 8/19, 9/19, 10/19, 11/19, 12/19, 13/19, 14/19, 15/19, 16/19, 17/19, 18/19], [1/19, 2/19, 3/19, 4/19, 5/19, 6/19, 7/19, 8/19, 9/19, 10/19, 11/19, 12/19, 13/19, 14/19, 15/19, 16/19, 17/19, 18/19], [1/19, 2/19, 3/19, 4/19, 5/19, 6/19, 7/19, 8/19, 9/19, 10/19, 11/19, 12/19, 13/19, 14/19, 15/19, 16/19, 17/19, 18/19], [1/19, 2/19, 3/19, 4/19, 5/19, 6/19, 7/19, 8/19, 9/19, 10/19, 11/19, 12/19, 13/19, 14/19, 15/19, 16/19, 17/19, 18/19], [1/19, 2/19, 3/19, 4/19, 5/19, 6/19, 7/19, 8/19, 9/19, 10/19, 11/19, 12/19, 13/19, 14/19, 15/19, 16/19, 17/19, 18/19], [1/19, 2/19, 3/19, 4/19, 5/19, 6/19, 7/19, 8/19, 9/19, 10/19, 11/19, 12/19, 13/19, 14/19, 15/19, 16/19, 17/19, 18/19], [1/19, 2/19, 3/19, 4/19, 5/19, 6/19, 7/19, 8/19, 9/19, 10/19, 11/19, 12/19, 13/19, 14/19, 15/19, 16/19, 17/19, 18/19]]
# not always 19, look at the second and third entries of the first output!


# Rediscover
# The fact that multiplication of numbers is commutative by running ALnm(X,1,2,10);
# -X[[1, 2]] + X[[2, 1]]

# That there is no Amitsur-Levitsky relation between three 2x2 matrices by running ALnm(X,2,3,10);
# 0

# The Amitsur-Levitsky relation between four 2x2 matrices by running ALnm(X,2,4,10);
# X[[1, 2, 3, 4]] - X[[1, 2, 4, 3]] - X[[1, 3, 2, 4]] + X[[1, 3, 4, 2]] + X[[1, 4, 2, 3]] - X[[1, 4, 3, 2]] - X[[2, 1, 3, 4]] + X[[2, 1, 4, 3]] + X[[2, 3, 1, 4]] - X[[2, 3, 4, 1]] - X[[2, 4, 1, 3]] + X[[2, 4, 3, 1]] + X[[3, 1, 2, 4]] - X[[3, 1, 4, 2]] - X[[3, 2, 1, 4]] + X[[3, 2, 4, 1]] + X[[3, 4, 1, 2]] - X[[3, 4, 2, 1]] - X[[4, 1, 2, 3]] + X[[4, 1, 3, 2]] + X[[4, 2, 1, 3]] - X[[4, 2, 3, 1]] - X[[4, 3, 1, 2]] + X[[4, 3, 2, 1]]

# That there is no Amitsur-Levitsky relation between five 3x3 matrices by running ALnm(X,3,5,10);
# 0

# The Amitsur-Levitsky relation between four 3x3 matrices by running ALnm(X,3,6,10);

# -X[[1, 2, 3, 4, 5, 6]] + X[[1, 2, 3, 4, 6, 5]] + X[[1, 2, 3, 5, 4, 6]] - X[[1, 2, 3, 5, 6, 4]] - X[[1, 2, 3, 6, 4, 5]] + X[[1, 2, 3, 6, 5, 4]] + X[[1, 2, 4, 3, 5, 6]] - X[[1, 2, 4, 3, 6, 5]] - X[[1, 2, 4, 5, 3, 6]] + X[[1, 2, 4, 5, 6, 3]] + X[[1, 2, 4, 6, 3, 5]] - X[[1, 2, 4, 6, 5, 3]] - X[[1, 2, 5, 3, 4, 6]] + X[[1, 2, 5, 3, 6, 4]] + X[[1, 2, 5, 4, 3, 6]] - X[[1, 2, 5, 4, 6, 3]] - X[[1, 2, 5, 6, 3, 4]] + X[[1, 2, 5, 6, 4, 3]] + X[[1, 2, 6, 3, 4, 5]] - X[[1, 2, 6, 3, 5, 4]] - X[[1, 2, 6, 4, 3, 5]] + X[[1, 2, 6, 4, 5, 3]] + X[[1, 2, 6, 5, 3, 4]] - X[[1, 2, 6, 5, 4, 3]] + X[[1, 3, 2, 4, 5, 6]] - X[[1, 3, 2, 4, 6, 5]] - X[[1, 3, 2, 5, 4, 6]] + X[[1, 3, 2, 5, 6, 4]] + X[[1, 3, 2, 6, 4, 5]] - X[[1, 3, 2, 6, 5, 4]] - X[[1, 3, 4, 2, 5, 6]] + X[[1, 3, 4, 2, 6, 5]] + X[[1, 3, 4, 5, 2, 6]] - X[[1, 3, 4, 5, 6, 2]] - X[[1, 3, 4, 6, 2, 5]] + X[[1, 3, 4, 6, 5, 2]] + X[[1, 3, 5, 2, 4, 6]] - X[[1, 3, 5, 2, 6, 4]] - X[[1, 3, 5, 4, 2, 6]] + X[[1, 3, 5, 4, 6, 2]] + X[[1, 3, 5, 6, 2, 4]] - X[[1, 3, 5, 6, 4, 2]] - X[[1, 3, 6, 2, 4, 5]] + X[[1, 3, 6, 2, 5, 4]] + X[[1, 3, 6, 4, 2, 5]] - X[[1, 3, 6, 4, 5, 2]] - X[[1, 3, 6, 5, 2, 4]] + X[[1, 3, 6, 5, 4, 2]] - X[[1, 4, 2, 3, 5, 6]] + X[[1, 4, 2, 3, 6, 5]] + X[[1, 4, 2, 5, 3, 6]] - X[[1, 4, 2, 5, 6, 3]] - X[[1, 4, 2, 6, 3, 5]] + X[[1, 4, 2, 6, 5, 3]] + X[[1, 4, 3, 2, 5, 6]] - X[[1, 4, 3, 2, 6, 5]] - X[[1, 4, 3, 5, 2, 6]] + X[[1, 4, 3, 5, 6, 2]] + X[[1, 4, 3, 6, 2, 5]] - X[[1, 4, 3, 6, 5, 2]] - X[[1, 4, 5, 2, 3, 6]] + X[[1, 4, 5, 2, 6, 3]] + X[[1, 4, 5, 3, 2, 6]] - X[[1, 4, 5, 3, 6, 2]] - X[[1, 4, 5, 6, 2, 3]] + X[[1, 4, 5, 6, 3, 2]] + X[[1, 4, 6, 2, 3, 5]] - X[[1, 4, 6, 2, 5, 3]] - X[[1, 4, 6, 3, 2, 5]] + X[[1, 4, 6, 3, 5, 2]] + X[[1, 4, 6, 5, 2, 3]] - X[[1, 4, 6, 5, 3, 2]] + X[[1, 5, 2, 3, 4, 6]] - X[[1, 5, 2, 3, 6, 4]] - X[[1, 5, 2, 4, 3, 6]] + X[[1, 5, 2, 4, 6, 3]] + X[[1, 5, 2, 6, 3, 4]] - X[[1, 5, 2, 6, 4, 3]] - X[[1, 5, 3, 2, 4, 6]] + X[[1, 5, 3, 2, 6, 4]] + X[[1, 5, 3, 4, 2, 6]] - X[[1, 5, 3, 4, 6, 2]] - X[[1, 5, 3, 6, 2, 4]] + X[[1, 5, 3, 6, 4, 2]] + X[[1, 5, 4, 2, 3, 6]] - X[[1, 5, 4, 2, 6, 3]] - X[[1, 5, 4, 3, 2, 6]] + X[[1, 5, 4, 3, 6, 2]] + X[[1, 5, 4, 6, 2, 3]] - X[[1, 5, 4, 6, 3, 2]] - X[[1, 5, 6, 2, 3, 4]] + X[[1, 5, 6, 2, 4, 3]] + X[[1, 5, 6, 3, 2, 4]] - X[[1, 5, 6, 3, 4, 2]] - X[[1, 5, 6, 4, 2, 3]] + X[[1, 5, 6, 4, 3, 2]] - X[[1, 6, 2, 3, 4, 5]] + X[[1, 6, 2, 3, 5, 4]] + X[[1, 6, 2, 4, 3, 5]] - X[[1, 6, 2, 4, 5, 3]] - X[[1, 6, 2, 5, 3, 4]] + X[[1, 6, 2, 5, 4, 3]] + X[[1, 6, 3, 2, 4, 5]] - X[[1, 6, 3, 2, 5, 4]] - X[[1, 6, 3, 4, 2, 5]] + X[[1, 6, 3, 4, 5, 2]] + X[[1, 6, 3, 5, 2, 4]] - X[[1, 6, 3, 5, 4, 2]] - X[[1, 6, 4, 2, 3, 5]] + X[[1, 6, 4, 2, 5, 3]] + X[[1, 6, 4, 3, 2, 5]] - X[[1, 6, 4, 3, 5, 2]] - X[[1, 6, 4, 5, 2, 3]] + X[[1, 6, 4, 5, 3, 2]] + X[[1, 6, 5, 2, 3, 4]] - X[[1, 6, 5, 2, 4, 3]] - X[[1, 6, 5, 3, 2, 4]] + X[[1, 6, 5, 3, 4, 2]] + X[[1, 6, 5, 4, 2, 3]] - X[[1, 6, 5, 4, 3, 2]] + X[[2, 1, 3, 4, 5, 6]] - X[[2, 1, 3, 4, 6, 5]] - X[[2, 1, 3, 5, 4, 6]] + X[[2, 1, 3, 5, 6, 4]] + X[[2, 1, 3, 6, 4, 5]] - X[[2, 1, 3, 6, 5, 4]] - X[[2, 1, 4, 3, 5, 6]] + X[[2, 1, 4, 3, 6, 5]] + X[[2, 1, 4, 5, 3, 6]] - X[[2, 1, 4, 5, 6, 3]] - X[[2, 1, 4, 6, 3, 5]] + X[[2, 1, 4, 6, 5, 3]] + X[[2, 1, 5, 3, 4, 6]] - X[[2, 1, 5, 3, 6, 4]] - X[[2, 1, 5, 4, 3, 6]] + X[[2, 1, 5, 4, 6, 3]] + X[[2, 1, 5, 6, 3, 4]] - X[[2, 1, 5, 6, 4, 3]] - X[[2, 1, 6, 3, 4, 5]] + X[[2, 1, 6, 3, 5, 4]] + X[[2, 1, 6, 4, 3, 5]] - X[[2, 1, 6, 4, 5, 3]] - X[[2, 1, 6, 5, 3, 4]] + X[[2, 1, 6, 5, 4, 3]] - X[[2, 3, 1, 4, 5, 6]] + X[[2, 3, 1, 4, 6, 5]] + X[[2, 3, 1, 5, 4, 6]] - X[[2, 3, 1, 5, 6, 4]] - X[[2, 3, 1, 6, 4, 5]] + X[[2, 3, 1, 6, 5, 4]] + X[[2, 3, 4, 1, 5, 6]] - X[[2, 3, 4, 1, 6, 5]] - X[[2, 3, 4, 5, 1, 6]] + X[[2, 3, 4, 5, 6, 1]] + X[[2, 3, 4, 6, 1, 5]] - X[[2, 3, 4, 6, 5, 1]] - X[[2, 3, 5, 1, 4, 6]] + X[[2, 3, 5, 1, 6, 4]] + X[[2, 3, 5, 4, 1, 6]] - X[[2, 3, 5, 4, 6, 1]] - X[[2, 3, 5, 6, 1, 4]] + X[[2, 3, 5, 6, 4, 1]] + X[[2, 3, 6, 1, 4, 5]] - X[[2, 3, 6, 1, 5, 4]] - X[[2, 3, 6, 4, 1, 5]] + X[[2, 3, 6, 4, 5, 1]] + X[[2, 3, 6, 5, 1, 4]] - X[[2, 3, 6, 5, 4, 1]] + X[[2, 4, 1, 3, 5, 6]] - X[[2, 4, 1, 3, 6, 5]] - X[[2, 4, 1, 5, 3, 6]] + X[[2, 4, 1, 5, 6, 3]] + X[[2, 4, 1, 6, 3, 5]] - X[[2, 4, 1, 6, 5, 3]] - X[[2, 4, 3, 1, 5, 6]] + X[[2, 4, 3, 1, 6, 5]] + X[[2, 4, 3, 5, 1, 6]] - X[[2, 4, 3, 5, 6, 1]] - X[[2, 4, 3, 6, 1, 5]] + X[[2, 4, 3, 6, 5, 1]] + X[[2, 4, 5, 1, 3, 6]] - X[[2, 4, 5, 1, 6, 3]] - X[[2, 4, 5, 3, 1, 6]] + X[[2, 4, 5, 3, 6, 1]] + X[[2, 4, 5, 6, 1, 3]] - X[[2, 4, 5, 6, 3, 1]] - X[[2, 4, 6, 1, 3, 5]] + X[[2, 4, 6, 1, 5, 3]] + X[[2, 4, 6, 3, 1, 5]] - X[[2, 4, 6, 3, 5, 1]] - X[[2, 4, 6, 5, 1, 3]] + X[[2, 4, 6, 5, 3, 1]] - X[[2, 5, 1, 3, 4, 6]] + X[[2, 5, 1, 3, 6, 4]] + X[[2, 5, 1, 4, 3, 6]] - X[[2, 5, 1, 4, 6, 3]] - X[[2, 5, 1, 6, 3, 4]] + X[[2, 5, 1, 6, 4, 3]] + X[[2, 5, 3, 1, 4, 6]] - X[[2, 5, 3, 1, 6, 4]] - X[[2, 5, 3, 4, 1, 6]] + X[[2, 5, 3, 4, 6, 1]] + X[[2, 5, 3, 6, 1, 4]] - X[[2, 5, 3, 6, 4, 1]] - X[[2, 5, 4, 1, 3, 6]] + X[[2, 5, 4, 1, 6, 3]] + X[[2, 5, 4, 3, 1, 6]] - X[[2, 5, 4, 3, 6, 1]] - X[[2, 5, 4, 6, 1, 3]] + X[[2, 5, 4, 6, 3, 1]] + X[[2, 5, 6, 1, 3, 4]] - X[[2, 5, 6, 1, 4, 3]] - X[[2, 5, 6, 3, 1, 4]] + X[[2, 5, 6, 3, 4, 1]] + X[[2, 5, 6, 4, 1, 3]] - X[[2, 5, 6, 4, 3, 1]] + X[[2, 6, 1, 3, 4, 5]] - X[[2, 6, 1, 3, 5, 4]] - X[[2, 6, 1, 4, 3, 5]] + X[[2, 6, 1, 4, 5, 3]] + X[[2, 6, 1, 5, 3, 4]] - X[[2, 6, 1, 5, 4, 3]] - X[[2, 6, 3, 1, 4, 5]] + X[[2, 6, 3, 1, 5, 4]] + X[[2, 6, 3, 4, 1, 5]] - X[[2, 6, 3, 4, 5, 1]] - X[[2, 6, 3, 5, 1, 4]] + X[[2, 6, 3, 5, 4, 1]] + X[[2, 6, 4, 1, 3, 5]] - X[[2, 6, 4, 1, 5, 3]] - X[[2, 6, 4, 3, 1, 5]] + X[[2, 6, 4, 3, 5, 1]] + X[[2, 6, 4, 5, 1, 3]] - X[[2, 6, 4, 5, 3, 1]] - X[[2, 6, 5, 1, 3, 4]] + X[[2, 6, 5, 1, 4, 3]] + X[[2, 6, 5, 3, 1, 4]] - X[[2, 6, 5, 3, 4, 1]] - X[[2, 6, 5, 4, 1, 3]] + X[[2, 6, 5, 4, 3, 1]] - X[[3, 1, 2, 4, 5, 6]] + X[[3, 1, 2, 4, 6, 5]] + X[[3, 1, 2, 5, 4, 6]] - X[[3, 1, 2, 5, 6, 4]] - X[[3, 1, 2, 6, 4, 5]] + X[[3, 1, 2, 6, 5, 4]] + X[[3, 1, 4, 2, 5, 6]] - X[[3, 1, 4, 2, 6, 5]] - X[[3, 1, 4, 5, 2, 6]] + X[[3, 1, 4, 5, 6, 2]] + X[[3, 1, 4, 6, 2, 5]] - X[[3, 1, 4, 6, 5, 2]] - X[[3, 1, 5, 2, 4, 6]] + X[[3, 1, 5, 2, 6, 4]] + X[[3, 1, 5, 4, 2, 6]] - X[[3, 1, 5, 4, 6, 2]] - X[[3, 1, 5, 6, 2, 4]] + X[[3, 1, 5, 6, 4, 2]] + X[[3, 1, 6, 2, 4, 5]] - X[[3, 1, 6, 2, 5, 4]] - X[[3, 1, 6, 4, 2, 5]] + X[[3, 1, 6, 4, 5, 2]] + X[[3, 1, 6, 5, 2, 4]] - X[[3, 1, 6, 5, 4, 2]] + X[[3, 2, 1, 4, 5, 6]] - X[[3, 2, 1, 4, 6, 5]] - X[[3, 2, 1, 5, 4, 6]] + X[[3, 2, 1, 5, 6, 4]] + X[[3, 2, 1, 6, 4, 5]] - X[[3, 2, 1, 6, 5, 4]] - X[[3, 2, 4, 1, 5, 6]] + X[[3, 2, 4, 1, 6, 5]] + X[[3, 2, 4, 5, 1, 6]] - X[[3, 2, 4, 5, 6, 1]] - X[[3, 2, 4, 6, 1, 5]] + X[[3, 2, 4, 6, 5, 1]] + X[[3, 2, 5, 1, 4, 6]] - X[[3, 2, 5, 1, 6, 4]] - X[[3, 2, 5, 4, 1, 6]] + X[[3, 2, 5, 4, 6, 1]] + X[[3, 2, 5, 6, 1, 4]] - X[[3, 2, 5, 6, 4, 1]] - X[[3, 2, 6, 1, 4, 5]] + X[[3, 2, 6, 1, 5, 4]] + X[[3, 2, 6, 4, 1, 5]] - X[[3, 2, 6, 4, 5, 1]] - X[[3, 2, 6, 5, 1, 4]] + X[[3, 2, 6, 5, 4, 1]] - X[[3, 4, 1, 2, 5, 6]] + X[[3, 4, 1, 2, 6, 5]] + X[[3, 4, 1, 5, 2, 6]] - X[[3, 4, 1, 5, 6, 2]] - X[[3, 4, 1, 6, 2, 5]] + X[[3, 4, 1, 6, 5, 2]] + X[[3, 4, 2, 1, 5, 6]] - X[[3, 4, 2, 1, 6, 5]] - X[[3, 4, 2, 5, 1, 6]] + X[[3, 4, 2, 5, 6, 1]] + X[[3, 4, 2, 6, 1, 5]] - X[[3, 4, 2, 6, 5, 1]] - X[[3, 4, 5, 1, 2, 6]] + X[[3, 4, 5, 1, 6, 2]] + X[[3, 4, 5, 2, 1, 6]] - X[[3, 4, 5, 2, 6, 1]] - X[[3, 4, 5, 6, 1, 2]] + X[[3, 4, 5, 6, 2, 1]] + X[[3, 4, 6, 1, 2, 5]] - X[[3, 4, 6, 1, 5, 2]] - X[[3, 4, 6, 2, 1, 5]] + X[[3, 4, 6, 2, 5, 1]] + X[[3, 4, 6, 5, 1, 2]] - X[[3, 4, 6, 5, 2, 1]] + X[[3, 5, 1, 2, 4, 6]] - X[[3, 5, 1, 2, 6, 4]] - X[[3, 5, 1, 4, 2, 6]] + X[[3, 5, 1, 4, 6, 2]] + X[[3, 5, 1, 6, 2, 4]] - X[[3, 5, 1, 6, 4, 2]] - X[[3, 5, 2, 1, 4, 6]] + X[[3, 5, 2, 1, 6, 4]] + X[[3, 5, 2, 4, 1, 6]] - X[[3, 5, 2, 4, 6, 1]] - X[[3, 5, 2, 6, 1, 4]] + X[[3, 5, 2, 6, 4, 1]] + X[[3, 5, 4, 1, 2, 6]] - X[[3, 5, 4, 1, 6, 2]] - X[[3, 5, 4, 2, 1, 6]] + X[[3, 5, 4, 2, 6, 1]] + X[[3, 5, 4, 6, 1, 2]] - X[[3, 5, 4, 6, 2, 1]] - X[[3, 5, 6, 1, 2, 4]] + X[[3, 5, 6, 1, 4, 2]] + X[[3, 5, 6, 2, 1, 4]] - X[[3, 5, 6, 2, 4, 1]] - X[[3, 5, 6, 4, 1, 2]] + X[[3, 5, 6, 4, 2, 1]] - X[[3, 6, 1, 2, 4, 5]] + X[[3, 6, 1, 2, 5, 4]] + X[[3, 6, 1, 4, 2, 5]] - X[[3, 6, 1, 4, 5, 2]] - X[[3, 6, 1, 5, 2, 4]] + X[[3, 6, 1, 5, 4, 2]] + X[[3, 6, 2, 1, 4, 5]] - X[[3, 6, 2, 1, 5, 4]] - X[[3, 6, 2, 4, 1, 5]] + X[[3, 6, 2, 4, 5, 1]] + X[[3, 6, 2, 5, 1, 4]] - X[[3, 6, 2, 5, 4, 1]] - X[[3, 6, 4, 1, 2, 5]] + X[[3, 6, 4, 1, 5, 2]] + X[[3, 6, 4, 2, 1, 5]] - X[[3, 6, 4, 2, 5, 1]] - X[[3, 6, 4, 5, 1, 2]] + X[[3, 6, 4, 5, 2, 1]] + X[[3, 6, 5, 1, 2, 4]] - X[[3, 6, 5, 1, 4, 2]] - X[[3, 6, 5, 2, 1, 4]] + X[[3, 6, 5, 2, 4, 1]] + X[[3, 6, 5, 4, 1, 2]] - X[[3, 6, 5, 4, 2, 1]] + X[[4, 1, 2, 3, 5, 6]] - X[[4, 1, 2, 3, 6, 5]] - X[[4, 1, 2, 5, 3, 6]] + X[[4, 1, 2, 5, 6, 3]] + X[[4, 1, 2, 6, 3, 5]] - X[[4, 1, 2, 6, 5, 3]] - X[[4, 1, 3, 2, 5, 6]] + X[[4, 1, 3, 2, 6, 5]] + X[[4, 1, 3, 5, 2, 6]] - X[[4, 1, 3, 5, 6, 2]] - X[[4, 1, 3, 6, 2, 5]] + X[[4, 1, 3, 6, 5, 2]] + X[[4, 1, 5, 2, 3, 6]] - X[[4, 1, 5, 2, 6, 3]] - X[[4, 1, 5, 3, 2, 6]] + X[[4, 1, 5, 3, 6, 2]] + X[[4, 1, 5, 6, 2, 3]] - X[[4, 1, 5, 6, 3, 2]] - X[[4, 1, 6, 2, 3, 5]] + X[[4, 1, 6, 2, 5, 3]] + X[[4, 1, 6, 3, 2, 5]] - X[[4, 1, 6, 3, 5, 2]] - X[[4, 1, 6, 5, 2, 3]] + X[[4, 1, 6, 5, 3, 2]] - X[[4, 2, 1, 3, 5, 6]] + X[[4, 2, 1, 3, 6, 5]] + X[[4, 2, 1, 5, 3, 6]] - X[[4, 2, 1, 5, 6, 3]] - X[[4, 2, 1, 6, 3, 5]] + X[[4, 2, 1, 6, 5, 3]] + X[[4, 2, 3, 1, 5, 6]] - X[[4, 2, 3, 1, 6, 5]] - X[[4, 2, 3, 5, 1, 6]] + X[[4, 2, 3, 5, 6, 1]] + X[[4, 2, 3, 6, 1, 5]] - X[[4, 2, 3, 6, 5, 1]] - X[[4, 2, 5, 1, 3, 6]] + X[[4, 2, 5, 1, 6, 3]] + X[[4, 2, 5, 3, 1, 6]] - X[[4, 2, 5, 3, 6, 1]] - X[[4, 2, 5, 6, 1, 3]] + X[[4, 2, 5, 6, 3, 1]] + X[[4, 2, 6, 1, 3, 5]] - X[[4, 2, 6, 1, 5, 3]] - X[[4, 2, 6, 3, 1, 5]] + X[[4, 2, 6, 3, 5, 1]] + X[[4, 2, 6, 5, 1, 3]] - X[[4, 2, 6, 5, 3, 1]] + X[[4, 3, 1, 2, 5, 6]] - X[[4, 3, 1, 2, 6, 5]] - X[[4, 3, 1, 5, 2, 6]] + X[[4, 3, 1, 5, 6, 2]] + X[[4, 3, 1, 6, 2, 5]] - X[[4, 3, 1, 6, 5, 2]] - X[[4, 3, 2, 1, 5, 6]] + X[[4, 3, 2, 1, 6, 5]] + X[[4, 3, 2, 5, 1, 6]] - X[[4, 3, 2, 5, 6, 1]] - X[[4, 3, 2, 6, 1, 5]] + X[[4, 3, 2, 6, 5, 1]] + X[[4, 3, 5, 1, 2, 6]] - X[[4, 3, 5, 1, 6, 2]] - X[[4, 3, 5, 2, 1, 6]] + X[[4, 3, 5, 2, 6, 1]] + X[[4, 3, 5, 6, 1, 2]] - X[[4, 3, 5, 6, 2, 1]] - X[[4, 3, 6, 1, 2, 5]] + X[[4, 3, 6, 1, 5, 2]] + X[[4, 3, 6, 2, 1, 5]] - X[[4, 3, 6, 2, 5, 1]] - X[[4, 3, 6, 5, 1, 2]] + X[[4, 3, 6, 5, 2, 1]] - X[[4, 5, 1, 2, 3, 6]] + X[[4, 5, 1, 2, 6, 3]] + X[[4, 5, 1, 3, 2, 6]] - X[[4, 5, 1, 3, 6, 2]] - X[[4, 5, 1, 6, 2, 3]] + X[[4, 5, 1, 6, 3, 2]] + X[[4, 5, 2, 1, 3, 6]] - X[[4, 5, 2, 1, 6, 3]] - X[[4, 5, 2, 3, 1, 6]] + X[[4, 5, 2, 3, 6, 1]] + X[[4, 5, 2, 6, 1, 3]] - X[[4, 5, 2, 6, 3, 1]] - X[[4, 5, 3, 1, 2, 6]] + X[[4, 5, 3, 1, 6, 2]] + X[[4, 5, 3, 2, 1, 6]] - X[[4, 5, 3, 2, 6, 1]] - X[[4, 5, 3, 6, 1, 2]] + X[[4, 5, 3, 6, 2, 1]] + X[[4, 5, 6, 1, 2, 3]] - X[[4, 5, 6, 1, 3, 2]] - X[[4, 5, 6, 2, 1, 3]] + X[[4, 5, 6, 2, 3, 1]] + X[[4, 5, 6, 3, 1, 2]] - X[[4, 5, 6, 3, 2, 1]] + X[[4, 6, 1, 2, 3, 5]] - X[[4, 6, 1, 2, 5, 3]] - X[[4, 6, 1, 3, 2, 5]] + X[[4, 6, 1, 3, 5, 2]] + X[[4, 6, 1, 5, 2, 3]] - X[[4, 6, 1, 5, 3, 2]] - X[[4, 6, 2, 1, 3, 5]] + X[[4, 6, 2, 1, 5, 3]] + X[[4, 6, 2, 3, 1, 5]] - X[[4, 6, 2, 3, 5, 1]] - X[[4, 6, 2, 5, 1, 3]] + X[[4, 6, 2, 5, 3, 1]] + X[[4, 6, 3, 1, 2, 5]] - X[[4, 6, 3, 1, 5, 2]] - X[[4, 6, 3, 2, 1, 5]] + X[[4, 6, 3, 2, 5, 1]] + X[[4, 6, 3, 5, 1, 2]] - X[[4, 6, 3, 5, 2, 1]] - X[[4, 6, 5, 1, 2, 3]] + X[[4, 6, 5, 1, 3, 2]] + X[[4, 6, 5, 2, 1, 3]] - X[[4, 6, 5, 2, 3, 1]] - X[[4, 6, 5, 3, 1, 2]] + X[[4, 6, 5, 3, 2, 1]] - X[[5, 1, 2, 3, 4, 6]] + X[[5, 1, 2, 3, 6, 4]] + X[[5, 1, 2, 4, 3, 6]] - X[[5, 1, 2, 4, 6, 3]] - X[[5, 1, 2, 6, 3, 4]] + X[[5, 1, 2, 6, 4, 3]] + X[[5, 1, 3, 2, 4, 6]] - X[[5, 1, 3, 2, 6, 4]] - X[[5, 1, 3, 4, 2, 6]] + X[[5, 1, 3, 4, 6, 2]] + X[[5, 1, 3, 6, 2, 4]] - X[[5, 1, 3, 6, 4, 2]] - X[[5, 1, 4, 2, 3, 6]] + X[[5, 1, 4, 2, 6, 3]] + X[[5, 1, 4, 3, 2, 6]] - X[[5, 1, 4, 3, 6, 2]] - X[[5, 1, 4, 6, 2, 3]] + X[[5, 1, 4, 6, 3, 2]] + X[[5, 1, 6, 2, 3, 4]] - X[[5, 1, 6, 2, 4, 3]] - X[[5, 1, 6, 3, 2, 4]] + X[[5, 1, 6, 3, 4, 2]] + X[[5, 1, 6, 4, 2, 3]] - X[[5, 1, 6, 4, 3, 2]] + X[[5, 2, 1, 3, 4, 6]] - X[[5, 2, 1, 3, 6, 4]] - X[[5, 2, 1, 4, 3, 6]] + X[[5, 2, 1, 4, 6, 3]] + X[[5, 2, 1, 6, 3, 4]] - X[[5, 2, 1, 6, 4, 3]] - X[[5, 2, 3, 1, 4, 6]] + X[[5, 2, 3, 1, 6, 4]] + X[[5, 2, 3, 4, 1, 6]] - X[[5, 2, 3, 4, 6, 1]] - X[[5, 2, 3, 6, 1, 4]] + X[[5, 2, 3, 6, 4, 1]] + X[[5, 2, 4, 1, 3, 6]] - X[[5, 2, 4, 1, 6, 3]] - X[[5, 2, 4, 3, 1, 6]] + X[[5, 2, 4, 3, 6, 1]] + X[[5, 2, 4, 6, 1, 3]] - X[[5, 2, 4, 6, 3, 1]] - X[[5, 2, 6, 1, 3, 4]] + X[[5, 2, 6, 1, 4, 3]] + X[[5, 2, 6, 3, 1, 4]] - X[[5, 2, 6, 3, 4, 1]] - X[[5, 2, 6, 4, 1, 3]] + X[[5, 2, 6, 4, 3, 1]] - X[[5, 3, 1, 2, 4, 6]] + X[[5, 3, 1, 2, 6, 4]] + X[[5, 3, 1, 4, 2, 6]] - X[[5, 3, 1, 4, 6, 2]] - X[[5, 3, 1, 6, 2, 4]] + X[[5, 3, 1, 6, 4, 2]] + X[[5, 3, 2, 1, 4, 6]] - X[[5, 3, 2, 1, 6, 4]] - X[[5, 3, 2, 4, 1, 6]] + X[[5, 3, 2, 4, 6, 1]] + X[[5, 3, 2, 6, 1, 4]] - X[[5, 3, 2, 6, 4, 1]] - X[[5, 3, 4, 1, 2, 6]] + X[[5, 3, 4, 1, 6, 2]] + X[[5, 3, 4, 2, 1, 6]] - X[[5, 3, 4, 2, 6, 1]] - X[[5, 3, 4, 6, 1, 2]] + X[[5, 3, 4, 6, 2, 1]] + X[[5, 3, 6, 1, 2, 4]] - X[[5, 3, 6, 1, 4, 2]] - X[[5, 3, 6, 2, 1, 4]] + X[[5, 3, 6, 2, 4, 1]] + X[[5, 3, 6, 4, 1, 2]] - X[[5, 3, 6, 4, 2, 1]] + X[[5, 4, 1, 2, 3, 6]] - X[[5, 4, 1, 2, 6, 3]] - X[[5, 4, 1, 3, 2, 6]] + X[[5, 4, 1, 3, 6, 2]] + X[[5, 4, 1, 6, 2, 3]] - X[[5, 4, 1, 6, 3, 2]] - X[[5, 4, 2, 1, 3, 6]] + X[[5, 4, 2, 1, 6, 3]] + X[[5, 4, 2, 3, 1, 6]] - X[[5, 4, 2, 3, 6, 1]] - X[[5, 4, 2, 6, 1, 3]] + X[[5, 4, 2, 6, 3, 1]] + X[[5, 4, 3, 1, 2, 6]] - X[[5, 4, 3, 1, 6, 2]] - X[[5, 4, 3, 2, 1, 6]] + X[[5, 4, 3, 2, 6, 1]] + X[[5, 4, 3, 6, 1, 2]] - X[[5, 4, 3, 6, 2, 1]] - X[[5, 4, 6, 1, 2, 3]] + X[[5, 4, 6, 1, 3, 2]] + X[[5, 4, 6, 2, 1, 3]] - X[[5, 4, 6, 2, 3, 1]] - X[[5, 4, 6, 3, 1, 2]] + X[[5, 4, 6, 3, 2, 1]] - X[[5, 6, 1, 2, 3, 4]] + X[[5, 6, 1, 2, 4, 3]] + X[[5, 6, 1, 3, 2, 4]] - X[[5, 6, 1, 3, 4, 2]] - X[[5, 6, 1, 4, 2, 3]] + X[[5, 6, 1, 4, 3, 2]] + X[[5, 6, 2, 1, 3, 4]] - X[[5, 6, 2, 1, 4, 3]] - X[[5, 6, 2, 3, 1, 4]] + X[[5, 6, 2, 3, 4, 1]] + X[[5, 6, 2, 4, 1, 3]] - X[[5, 6, 2, 4, 3, 1]] - X[[5, 6, 3, 1, 2, 4]] + X[[5, 6, 3, 1, 4, 2]] + X[[5, 6, 3, 2, 1, 4]] - X[[5, 6, 3, 2, 4, 1]] - X[[5, 6, 3, 4, 1, 2]] + X[[5, 6, 3, 4, 2, 1]] + X[[5, 6, 4, 1, 2, 3]] - X[[5, 6, 4, 1, 3, 2]] - X[[5, 6, 4, 2, 1, 3]] + X[[5, 6, 4, 2, 3, 1]] + X[[5, 6, 4, 3, 1, 2]] - X[[5, 6, 4, 3, 2, 1]] + X[[6, 1, 2, 3, 4, 5]] - X[[6, 1, 2, 3, 5, 4]] - X[[6, 1, 2, 4, 3, 5]] + X[[6, 1, 2, 4, 5, 3]] + X[[6, 1, 2, 5, 3, 4]] - X[[6, 1, 2, 5, 4, 3]] - X[[6, 1, 3, 2, 4, 5]] + X[[6, 1, 3, 2, 5, 4]] + X[[6, 1, 3, 4, 2, 5]] - X[[6, 1, 3, 4, 5, 2]] - X[[6, 1, 3, 5, 2, 4]] + X[[6, 1, 3, 5, 4, 2]] + X[[6, 1, 4, 2, 3, 5]] - X[[6, 1, 4, 2, 5, 3]] - X[[6, 1, 4, 3, 2, 5]] + X[[6, 1, 4, 3, 5, 2]] + X[[6, 1, 4, 5, 2, 3]] - X[[6, 1, 4, 5, 3, 2]] - X[[6, 1, 5, 2, 3, 4]] + X[[6, 1, 5, 2, 4, 3]] + X[[6, 1, 5, 3, 2, 4]] - X[[6, 1, 5, 3, 4, 2]] - X[[6, 1, 5, 4, 2, 3]] + X[[6, 1, 5, 4, 3, 2]] - X[[6, 2, 1, 3, 4, 5]] + X[[6, 2, 1, 3, 5, 4]] + X[[6, 2, 1, 4, 3, 5]] - X[[6, 2, 1, 4, 5, 3]] - X[[6, 2, 1, 5, 3, 4]] + X[[6, 2, 1, 5, 4, 3]] + X[[6, 2, 3, 1, 4, 5]] - X[[6, 2, 3, 1, 5, 4]] - X[[6, 2, 3, 4, 1, 5]] + X[[6, 2, 3, 4, 5, 1]] + X[[6, 2, 3, 5, 1, 4]] - X[[6, 2, 3, 5, 4, 1]] - X[[6, 2, 4, 1, 3, 5]] + X[[6, 2, 4, 1, 5, 3]] + X[[6, 2, 4, 3, 1, 5]] - X[[6, 2, 4, 3, 5, 1]] - X[[6, 2, 4, 5, 1, 3]] + X[[6, 2, 4, 5, 3, 1]] + X[[6, 2, 5, 1, 3, 4]] - X[[6, 2, 5, 1, 4, 3]] - X[[6, 2, 5, 3, 1, 4]] + X[[6, 2, 5, 3, 4, 1]] + X[[6, 2, 5, 4, 1, 3]] - X[[6, 2, 5, 4, 3, 1]] + X[[6, 3, 1, 2, 4, 5]] - X[[6, 3, 1, 2, 5, 4]] - X[[6, 3, 1, 4, 2, 5]] + X[[6, 3, 1, 4, 5, 2]] + X[[6, 3, 1, 5, 2, 4]] - X[[6, 3, 1, 5, 4, 2]] - X[[6, 3, 2, 1, 4, 5]] + X[[6, 3, 2, 1, 5, 4]] + X[[6, 3, 2, 4, 1, 5]] - X[[6, 3, 2, 4, 5, 1]] - X[[6, 3, 2, 5, 1, 4]] + X[[6, 3, 2, 5, 4, 1]] + X[[6, 3, 4, 1, 2, 5]] - X[[6, 3, 4, 1, 5, 2]] - X[[6, 3, 4, 2, 1, 5]] + X[[6, 3, 4, 2, 5, 1]] + X[[6, 3, 4, 5, 1, 2]] - X[[6, 3, 4, 5, 2, 1]] - X[[6, 3, 5, 1, 2, 4]] + X[[6, 3, 5, 1, 4, 2]] + X[[6, 3, 5, 2, 1, 4]] - X[[6, 3, 5, 2, 4, 1]] - X[[6, 3, 5, 4, 1, 2]] + X[[6, 3, 5, 4, 2, 1]] - X[[6, 4, 1, 2, 3, 5]] + X[[6, 4, 1, 2, 5, 3]] + X[[6, 4, 1, 3, 2, 5]] - X[[6, 4, 1, 3, 5, 2]] - X[[6, 4, 1, 5, 2, 3]] + X[[6, 4, 1, 5, 3, 2]] + X[[6, 4, 2, 1, 3, 5]] - X[[6, 4, 2, 1, 5, 3]] - X[[6, 4, 2, 3, 1, 5]] + X[[6, 4, 2, 3, 5, 1]] + X[[6, 4, 2, 5, 1, 3]] - X[[6, 4, 2, 5, 3, 1]] - X[[6, 4, 3, 1, 2, 5]] + X[[6, 4, 3, 1, 5, 2]] + X[[6, 4, 3, 2, 1, 5]] - X[[6, 4, 3, 2, 5, 1]] - X[[6, 4, 3, 5, 1, 2]] + X[[6, 4, 3, 5, 2, 1]] + X[[6, 4, 5, 1, 2, 3]] - X[[6, 4, 5, 1, 3, 2]] - X[[6, 4, 5, 2, 1, 3]] + X[[6, 4, 5, 2, 3, 1]] + X[[6, 4, 5, 3, 1, 2]] - X[[6, 4, 5, 3, 2, 1]] + X[[6, 5, 1, 2, 3, 4]] - X[[6, 5, 1, 2, 4, 3]] - X[[6, 5, 1, 3, 2, 4]] + X[[6, 5, 1, 3, 4, 2]] + X[[6, 5, 1, 4, 2, 3]] - X[[6, 5, 1, 4, 3, 2]] - X[[6, 5, 2, 1, 3, 4]] + X[[6, 5, 2, 1, 4, 3]] + X[[6, 5, 2, 3, 1, 4]] - X[[6, 5, 2, 3, 4, 1]] - X[[6, 5, 2, 4, 1, 3]] + X[[6, 5, 2, 4, 3, 1]] + X[[6, 5, 3, 1, 2, 4]] - X[[6, 5, 3, 1, 4, 2]] - X[[6, 5, 3, 2, 1, 4]] + X[[6, 5, 3, 2, 4, 1]] + X[[6, 5, 3, 4, 1, 2]] - X[[6, 5, 3, 4, 2, 1]] - X[[6, 5, 4, 1, 2, 3]] + X[[6, 5, 4, 1, 3, 2]] + X[[6, 5, 4, 2, 1, 3]] - X[[6, 5, 4, 2, 3, 1]] - X[[6, 5, 4, 3, 1, 2]] + X[[6, 5, 4, 3, 2, 1]]
















### copied from C22.txt 

#C22.txt, April 14, 2025
Help22:=proc(): print(` BAc(f), Likelyc(q,r) , LikelycC(q,r) , ConfirmR(q,r). RM(n,K), ALnm(X,n,m,K)) `): end:
read `QC.txt`:
#BAc(f): a clever version of BA(f): the best relative approximation to the fraction f
#with denominator <=sqrt(denom(f))
BAc:=proc(f) local L,i:
L:=CVG(f):
for i from 1 to nops(L) while denom(L[i])^2<=denom(f) do od:
L[i-1]:
end:

#Likelyc(q,r): All c less than q such that |{rc}_q| <=r/2
Likelyc:=proc(q,r) local c,L:
L:=[]:

for c from 1 to q-1 do
  if abs(mods(r*c,q))<=r/2 then
    if c mod 2=0 then
      c:=c+1:
     fi:
    L:=[op(L),c]:
  fi:
od:

L:
end:

#LikelycC(q,r): All c less than q such that |{rc}_q| <=r/2 the clever way
#i.e. all integer multioles of q/r rounded


#LikelycC(q,r): All c less than q such that |{rc}_q| <=r/2
LikelycC:=proc(q,r) local i,L,c:
L:=[]:

#integer multiples of q/r
for i from 1 to r-1 do
   c:=round(q/r*i):
   if c mod 2=0 then
     c:=c+1:
    fi:
  
    L:=[op(L),c]:
 od:

L:
end:

#ConfirmR(q,r)
ConfirmR:=proc(q,r) local L,i:
if r^2>=q then
   RETURN(FAIL):
fi:
L:=LikelycC(q,r):

[seq(BAc(L[i]/q),i=1..nops(L))];
end:


#added after class
##start Amitsur-Levitstky (Joseph's project)

with(combinat):

#Mul(A,B): the product of matrix A and B (assuming that it exists). Try:
#Mul([[a1,a2]],[[b1],[b2]]);
Mul:=proc(A,B) local i,j,k,n:
[seq([seq(add(A[i][k]*B[k][j],k=1..nops(A[i])),j=1..nops(B[1]))],i=1..nops(A))]:
end:



#RM(n,K): a random n by n matrix with entries in [1,K]
RM:=proc(n,K) local ra,i,j:
  ra := rand(1..K):
[seq([seq(ra(),i=1..n)],j=1..n)]:
end:

#RMs(n,K,m): A list of m random nxn matrices with entries from [1,K]
RMs:=proc(n,K,m) local i:
  [seq(RM(n,K), i=1..m)]:
end:

#ALnm(X,n,m,K): Tries to find a relation between m  n by n matrices. Try:
#ALnm(X,1,2); Alnm(X,2,4); Alnm(X,3,6);
ALnm:=proc(X,n,m,K) local A, pi, eqs, M, var, i, var1, v, T,JK,M1,c,i1:
  var := {seq(c[pi], pi in permute(m))}:
JK:=add(c[pi]*X[pi],pi in permute(m)):

  eqs := {}:

  for i from 1 to trunc(m!/n^2)+5 do
    A := RMs(n,K,m):
   M:=[[0$n]$n]:
    for pi in permute(m) do
        M1:=A[pi[1]]:
    for i1 from 2 to m do
     M1:=Mul(M1,A[pi[i1]]):
    od:

    M:=expand(M+c[pi]*M1):
    od:
    eqs := eqs union {seq(op(M[i]),i=1..nops(M))}:
    
od:

  var := solve(eqs, var):
if subs(var,JK)=0 then
  RETURN(0):
fi:


  var1:={}:
  for v in var do
    if op(1,v)=op(2,v) then
      var1:=var1 union {op(1,v)}:
    fi:
  od:

subs(var1[1]=1,subs(var,JK)):
end: