#Below are the 1001 denominators which occur in the 27142 Krypto-5 formulas
#File is incomplete.

DENOMS:=[
x1, x2, x3, x4, x5,
seq(seq(op([x||i + x||j, x||i - x||j, x||i * x||j]), i=1..4), j=i+1..5),
#   30 of those, length=11, 2 vars
seq(seq(seq(op([x||i + x||j + x||k, x||i + x||j - x||k, x||i - x||j + x||k, 
x||i - x||j - x||k , x||i * x||j * x||k ]) ,i=1..3),j=i+1..4),k=j+1..5),
#   50 of those, length=16, 3 vars
seq(seq(seq(op([x||i + x||j * x||k, x||i * x||j + x||k, x||i * x||k + x||j, 
x||i - x||j * x||k , x||i * x||j - x||k, x||i * x||k - x||j ]) ,i=1..3),j=i+1..4),k=j+1..5),
#   60 of those, length=20, 3 vars
seq(seq(seq(op([x||i *(x||j + x||k), x||i *(x||j - x||k), x||j *(x||k + x||i), 
x||j*(x||k - x||i), x||k *(x||i + x||k),x||k *(x||i - x||k) ]) ,i=1..3),j=i+1..4),k=j+1..5),
#   60 of those, length=20, 3 vars
seq(seq(seq(seq(op([
x||i +x||j + x||k + x||l, x||i + x||j + x||k - x||l, x||i + x||j - x||k + x||l,
x||i -x||j + x||k + x||l, x||i + x||j - x||k - x||l, x||i - x||j + x||k - x||l,
x||i -x||j - x||k + x||l, x||i - x||j - x||k - x||l, x||i * x||j * x||k * x||l
]),i=1..2),j=i+1..3),k=j+1..4),l=k+1..5),
#   45 of those, length=21, 4 vars
1]:


#UNFINISHED SYMMETRIZATION! (5 times as many; bring in x5)
x3*x4-x1-x2
x2*x3-x1-x4
x1*x3-x2-x4
x3-x2-x1*x4
x3-x4-x1*x2
x3-x1-x2*x4
x2+x3*x4-x1
x4+x2*x3-x1
x4+x1*x3-x2
x3+x2*x4-x1
x3+x1*x4-x2
x3+x1*x2-x4
x1+x2*x3-x4
x2+x1*x3-x4
x1+x3*x4-x2
x1+x2+x3*x4
x1+x3+x2*x4
x1+x4+x2*x3
x1+x3-x2*x4
x2+x3+x1*x4
x2+x4+x1*x3
x3+x4-x1*x2
x2+x3-x1*x4
x3+x4+x1*x2

x2*x3*x4-x1
x3-x1*x2*x4
x1*x3*x4-x2
x1*x2*x3-x4
x1+x2*x3*x4
x4+x1*x2*x3
x3+x1*x2*x4
x2+x1*x3*x4

(x3+x4)*x1*x2
(x2+x4)*x1*x3
(x2+x3)*x1*x4
(x1+x4)*x2*x3
(x1+x3)*x2*x4
(x1+x2)*x3*x4

(x4-x2)*x1*x3
(x4-x1)*x2*x3
(x3-x4)*x1*x2
(x3-x2)*x1*x4
(x3-x1)*x2*x4
(x2-x1)*x3*x4

(x3-x2-x4)*x1
(x3-x1-x4)*x2
(x3-x1-x2)*x4
(x4-x1-x2)*x3
(x3+x4-x2)*x1
(x1+x3-x4)*x2
(x1+x3-x2)*x4
(x1+x4-x2)*x3
(x3+x4-x1)*x2
(x2+x4-x1)*x3
(x2+x3-x4)*x1
(x2+x3-x1)*x4
(x2+x3+x4)*x1
(x1+x2+x4)*x3
(x1+x3+x4)*x2
(x1+x2+x3)*x4
#   300 of those, length=25, 4 vars

x1*x4+x2*x3
x1*x3+x2*x4
x1*x2+x3*x4
x3*x4-x1*x2
x2*x3-x1*x4
x1*x3-x2*x4
(x2+x3*x4)*x1
(x2+x1*x4)*x3
(x2+x1*x3)*x4
(x1+x3*x4)*x2
(x1+x2*x4)*x3
(x1+x2*x3)*x4
(x3+x2*x4)*x1
(x3+x1*x4)*x2
(x3+x1*x2)*x4
(x4+x2*x3)*x1
(x4+x1*x3)*x2
(x4+x1*x2)*x3
(x1*x4-x2)*x3
(x1*x3-x4)*x2
(x1*x3-x2)*x4
(x2*x4-x1)*x3
(x2*x3-x4)*x1
(x2*x3-x1)*x4
(x3*x4-x2)*x1
(x3*x4-x1)*x2
(x1*x2-x4)*x3
(x2*x4-x3)*x1
(x1*x4-x3)*x2
(x1*x2-x3)*x4
(x1+x4)*(x2+x3)
(x1+x3)*(x2+x4)
(x1+x2)*(x3+x4)
(x2+x4)*(x3-x1)
(x2+x3)*(x4-x1)
(x1+x4)*(x3-x2)
(x1+x3)*(x4-x2)
(x1+x2)*(x3-x4)
(x3+x4)*(x2-x1)
(x3-x1)*(x4-x2)
(x2-x1)*(x3-x4)
(x3-x2)*(x4-x1)
#   210 of those, length=29, 4 vars

x4+x1*x2+x1*x3
x3+x1*x2+x1*x4
x4+x1*x2+x2*x3
x4+x1*x3+x2*x3
x2+x1*x3+x1*x4
x3+x1*x2+x2*x4
x3+x1*x4+x2*x4
x2+x1*x3+x3*x4
x2+x1*x4+x3*x4
x1+x2*x3+x2*x4
x1+x2*x3+x3*x4
x1+x2*x4+x3*x4

x1*x3+x3*x4-x2
x2*x3+x3*x4-x1
x1*x2+x2*x3-x4
x1*x3+x1*x4-x2
x1*x2+x1*x3-x4
x2*x3+x2*x4-x1
x1*x3+x2*x3-x4
x1*x4+x3*x4-x2
x2*x4+x3*x4-x1
x1*x2+x2*x4-x3
x1*x2+x1*x4-x3
x1*x4+x2*x4-x3

x2+x3*x4-x1*x3
x1+x3*x4-x2*x3
x4+x2*x3-x1*x2
x3+x1*x2-x1*x4
x2+x1*x3-x1*x4
x1+x2*x3-x2*x4
x2+x3*x4-x1*x4
x3+x2*x4-x1*x2
x3+x2*x4-x1*x4
x3+x1*x4-x1*x2
x1+x3*x4-x2*x4
x4+x1*x3-x1*x2
x4+x2*x3-x1*x3
x3+x1*x2-x2*x4
x3+x1*x4-x2*x4
x1+x2*x3-x3*x4
x2+x1*x3-x3*x4
x4+x1*x2-x2*x3
x2+x1*x4-x1*x3
x2+x1*x4-x3*x4
x1+x2*x4-x2*x3
x4+x1*x3-x2*x3
x1+x2*x4-x3*x4
x4+x1*x2-x1*x3

#   240 of those, length=34, 4 vars
]:


I plan to look again at the completely asymmetric Krypto-5 formulas

X75:=[
(x1*x3*x5+x1*x2+x3*x4)/(x4+x1*x5),
(x2*x3*x5-x1*x2*x5-x4)/(x3-x1)/x5,
(x2*x4-x1*x4-x5)/x3/x4,
(x2*x3-x1*x5)/x1/x2/x4,
(x2*x5+x3*x4*x5-x1*x3)/x5,
(x5+x1*x4)*x3/x2/x4,
-(x1-x2*x3*x5+x2*x3*x4)/(x4-x5)/x2,
-x1*x5/(-x1*x4+x1*x2+x3),
(-x1*x2-x4*x5+x1*x3*x5)/x1,
-(x1*x3+x1*x2*x4-x2*x5)/x1/x2,
-x4*x5/(x2+x3*x4-x1),
-(-x2*x5+x1*x5+x4)*x3/(x2-x1),
-(x3*x5-x2*x4+x1*x4)/x4/x5,
(x3*x4*x5-x1*x5-x2)/x3/x5,
-(x1+x2*x3*x4-x2*x5)/x2/x3,
(x1*x3+x2*x3*x4-x2*x5)/(x1+x2*x4),
-(x3*x4-x4*x5-x2)*x1,
x4/(x3+x2*x5-x1*x2),
(-x1*x4-x1*x2*x5+x2*x3)/(x4+x2*x5),
-(x2*x4-x1*x4-x5)/x3/x4,
(x3*x5-x1*x2)*x4/x2/x5,
x4*x5/(x2+x3*x5)/x1,
-x1*x2/(x3-x4-x1*x5),
-(x3*x4-x4*x5-x1)*x2/(x3-x5),
(x1*x3-x1*x5-x2)*x4,
-(x3*x4*x5-x1*x5-x2)/x3/x5,
(x4-x5)*(x1*x3-x2)/x3,
-(x2*x5+x1*x3*x5-x1*x4)/(x3*x5-x4),
-(x3-x5)*x1/(x3*x4-x4*x5-x2),
(x3*x4-x4*x5-x2)*x1/x4,
(x4*x5+x1*x3*x5-x1*x2)/x1,
-(x3*x4-x1*x3-x5)*x2/x3,
x1*x2/(x1*x3-x5)/x4,
(x2*x3-x1)/(x4-x5)/x3,
-(x2+x4*x5-x1)*x3/x5,
-(x4+x1*x3)/(x5-x2)/x1,
-(x2*x3+x3*x4*x5-x1*x4)/x3,
(x3-x1*x4*x5+x2*x4*x5)/x4,
(x4+x3*x5-x2*x3-x1)/x3,
(x3+x2*x4*x5-x1*x5)/x2/x5,
(x1*x2+x3*x5-x2*x4*x5)/x2,
x1+x2*x5-x2*x4-x3,
(x2+x4*x5-x1)*x3/x5,
-x1*x3/(x3*x5-x2)/x4,
-(x2+x1*x5-x1*x4-x3*x5+x3*x4)/(x3-x1),
(x3*x5+x1*x2*x3-x4)/x2/x3,
(-x2*x4+x3*x5-x1*x5)/x2,
(-x1*x4+x3*x5-x2*x3)/(x5-x2)/x4,
(x3-x4*x5)*x2/x1/x4,
(x1*x3-x2*x4+x2*x5)/x2/x3,
(x1*x3+x2*x5-x1*x2*x4)/(x1*x4-x5),
-(x3*x5+x1*x4-x1*x2)/(x4-x2)/x5,
-(x5+x1*x4-x2)/x1/x3,
(x3-x5)*x1/(x3*x4-x4*x5-x2),
(x4+x3*x5-x2)/x1/x5,
(x3*x5+x2*x4-x1*x4)/x5,
(x1+x2*x4-x2*x5)/(x4-x5)/x3,
(x1*x2*x3-x1*x2*x4-x5)/x1,
(x2+x1*x5)*(x3-x4)/x5,
(x1*x4*x5+x2*x5+x3*x4)/x4,
(x3-x2)*x4/(x4*x5-x1),
-(x4-x1*x2)*x5/x2/x3,
-(x3+x4*x5-x2*x5)/(x4-x2)/x1,
(x3*x4+x2*x4*x5-x1*x5)/(x2*x4-x1),
-(x3+x4*x5-x1*x4)/x2,
-x1/(x4+x3*x5-x2*x5),
(-x1*x5-x2*x4+x3*x4*x5)/(x3*x4-x1),
(x3+x1*x4-x1*x2)/x5,
-x1*x2*x5/(x2*x3-x4*x5),
(x3+x1*x4+x2*x4*x5)/x4/x5,
-(x1*x2+x2*x3*x4-x5)/x2/x4,
-(x1*x5+x2*x3*x5-x2*x4)/x2,
(x3*x4+x1*x2*x4-x1*x5)/x1/x4,
-(x3-x2)*x5/(x1+x4*x5),
x4*x5/(x3+x2*x4-x1*x4)
]:



#The set of 216 formulas (kept the original output for safekeeping...)

A:=[
(x3+x1*x4*x5+x2*x4*x5)/(x1+x2)/x4,
(x5-x1-x2)/(x3-x4),
(x1+x2+x3*x4-x5)/x4,
(x1*x4+x2*x4+x3-x4*x5)/(x5-x1-x2),
(x4-x1*x2*x5+x1*x2*x3)/x1/x2,
-(x1+x2)*(x3-x4*x5)/x5,
-(x5-x1*x2*x4+x1*x2*x3)/x1/x2,
-(x1*x3+x2*x3-x4)/(x1+x2)/x5,
-x3*x4/(x4*x5-x1*x2),
x5-x3*x4+x1*x3+x2*x3,
-x3/(x5-x1*x2)/x4,
-(x3+x4*x5-x1-x2)/x5,
x3*x4-x1*x2*x3-x5,
x3/(x4+x1*x2-x5),
-(-x5+x1*x3-x1*x4+x2*x3-x2*x4)/(x3-x4),
-x3*x4/(x1*x2*x3-x5),
-(x3*x4-x5)/(x1+x2)/x3,
-(x3*x4-x1*x2*x4-x5)/(x3-x1*x2),
(x4+x3*x5)/(x1+x2)/x5,
(x1+x2+x3*x5)*x4/x3,
(-x1*x4*x5-x2*x4*x5+x3)/(x1+x2)/x5,
(-x4*x5+x1*x2*x4+x3)/x4,
-(-x4+x5+x1*x3+x2*x3)/x3,
(x3+x1*x5+x2*x5)/x4,
(x4+x3*x5-x1*x5-x2*x5)/(x3-x1-x2),
(x3-x4)/(x1+x2)/x5,
x1*x2*x3/(x3*x5-x4),
(x3*x5-x1*x2*x4)/x1/x2/x5,
-(x1*x2*x3-x4)/x3/x5,
x3/(x1+x2)/(x4-x5),
(x3+x1*x2)/(x4-x5),
(x3*x4+x1*x2*x4-x5)/(x3+x1*x2),
-(x1+x2)*x3/(x3*x5-x4),
(x1+x2)*(x4-x5)/x3,
x1*x2*x3/(x4+x3*x5),
-(x3*x4-x4*x5-x1-x2)/(x3-x5),
-(x1*x3+x2*x3-x5)/x4,
(x4+x1*x2*x3)*x5/x1/x2,
(x1+x2)/(x3-x5)/x4,
-(x1*x4+x2*x4+x3-x4*x5)/(x5-x1-x2),
-(-x1*x3-x2*x3-x5+x3*x4)/x3,
x4*x5/(-x1*x5-x2*x5+x3),
-x5+x1*x3-x1*x4+x2*x3-x2*x4,
-(x4*x5-x1-x2)*x3/x5,
(x1+x2)*(x3-x4)*x5,
-(x1*x3+x2*x3-x5)*x4/(x1+x2),
(x4+x3*x5)/x1/x2/x5,
-(x1+x2)*x5/(x3-x4*x5),
(x3-x5-x1*x2*x4)/x1/x2,
x4/(x3+x1*x5+x2*x5),
-(x1+x2)*(x3*x4-x5)/x4,
(x3+x1*x4*x5+x2*x4*x5)/x5,
(x4+x1*x2)*x3/x5,
-(x5-x1*x2*x3+x3*x4)/(x4-x1*x2),
(x3+x1*x2-x4)*x5,
-(x3*x4-x1*x2*x5)/x3/x5,
(x5+x1*x3+x2*x3)*x4/(x1+x2),
(x1*x2+x4*x5)*x3/x4,
-(x3-x5)*x4/(x1+x2),
x3*x5/(x4+x1*x2*x5),
x3*x4+x1*x2*x4-x5,
-(x3*x4+x1*x2*x4-x5)/(x3+x1*x2),
-(-x1*x5-x2*x5+x3)/(x1+x2)/x4,
(x3+x1*x4-x1*x5+x2*x4-x2*x5)/(x1+x2),
-(x3-x5)/(x4-x1*x2),
(x3-x1-x2-x4*x5)/x5,
-(x4-x1*x2*x5)/x3/x5,
-(-x1*x5-x2*x5+x3*x4)/(x1+x2)/x3,
(x1*x3+x2*x3-x5)*x4,
(x4+x1*x2-x5)/x3,
(x1*x3*x4+x2*x3*x4-x5)/x3,
(x1*x2*x3-x4*x5)/x1/x2/x5,
(x3*x4-x4*x5-x1-x2)/x4,
-(x3-x1*x2)*x5/x4,
-x1*x4-x2*x4-x5+x3*x4,
(x3-x4*x5)*x1*x2/x4,
(x5+x1*x2*x4)/x3/x4,
x4*x5/(x3*x5-x1*x2),
(x1+x2)*(x4+x3*x5)/x3,
-x3/(-x1*x5-x2*x5+x4),
(x1*x4+x2*x4+x3*x5)/x4/x5,
(x3-x4)/(x5-x1-x2),
-(x3*x4-x1-x2)/x3/x5,
-(x3*x5-x4)/x1/x2/x3,
-(x3+x1*x2-x4*x5)/x5,
x3*x5/(x1*x2+x4*x5),
(x4*x5-x1*x2)/x3/x5,
-(x1+x2)*x5/(x1*x3+x2*x3-x4),
(x4+x3*x5)*x1*x2/x3,
(x1*x2+x3*x4-x4*x5)/x4,
(x4*x5-x1-x2)*x3/x5,
-(x3*x5-x1*x2)*x4/x3,
x3/(x5+x1*x2)/x4,
x5+x1*x3-x1*x4+x2*x3-x2*x4,
(x4+x1*x2*x3)/x1/x2/x5,
(x1+x2)*x3/(x4+x1*x5+x2*x5),
(x3*x5-x1*x2)*x4/x5,
x3-x5-x1*x4-x2*x4,
-(-x1*x5-x2*x5+x4)/x3/x5,
(x5+x1*x4+x2*x4)*x3/x4,
(x3-x1-x2)*x4/x5,
-(x3-x4*x5)/x1/x2/x5,
(x4+x3*x5-x1*x5-x2*x5)/x5,
-(x3*x4-x1*x2)/x3/x5,
x5-x1*x2*x3+x3*x4,
(x3-x1*x2*x4+x1*x2*x5)/(x4-x5),
(x5-x1-x2)*(x3-x4),
(-x1*x5-x2*x5+x4)/x3/x5,
(x4+x1*x2+x3*x5)/x3,
x1*x2*x4/(x3-x4*x5),
-(x4+x1*x2*x3-x5)/x3,
-(x3-x4-x1*x5-x2*x5)/(x1+x2),
(x3+x1*x2-x4*x5)/x5,
(x3-x1*x2)*x4/x5,
(x3*x5-x4)/(x1+x2)/x5,
-(x1*x2+x3*x5-x4*x5)/x5,
(x5+x1*x2*x3-x4)/x3,
(-x1*x4*x5-x2*x4*x5+x3)/x5,
(x1+x2+x3*x4-x3*x5)/(x4-x5),
(x1*x2*x3-x4)/x1/x2/x5,
(x4+x3*x5-x1*x2)/x5,
(x3-x5)*(x4-x1*x2),
-(x3+x1*x2-x5)/x4,
-(x3*x4-x4*x5-x1-x2)/x4,
x3*x5/(x3*x4-x1-x2),
-(-x1*x4-x2*x4+x3*x5)/x4/x5,
x3/(x4-x1*x2)/x5,
x1*x2*x3/(x5+x1*x2*x4),
(x3-x1*x2*x5+x1*x2*x4)/(x4-x5),
x1*x2*x3-x1*x2*x5-x4,
-(-x1*x4-x2*x4+x3)/x5,
-x3/(x4-x5)/x1/x2,
x4*x5-x3*x5-x1*x2,
x1*x2/(x3-x5)/x4,
(x1+x2)*x3/(x4+x3*x5),
-(x4-x1-x2)*x5/x3,
(-x1*x5-x2*x5+x3*x4)/(x1+x2)/x3,
-(x3-x5-x1*x2*x4)/x1/x2,
-(x4-x1*x2*x3+x3*x5)/x3,
(x3+x1*x4+x2*x4)/x4/x5,
(x4+x3*x5+x1*x2*x5)/x5,
x1*x2+x3*x5-x4*x5,
-(x1+x2)*x3/(-x1*x5-x2*x5+x4),
-(x3-x5)*x1*x2/x4,
(x3-x1*x2-x4*x5)/x5,
x4+x3*x5+x1*x2*x5,
-x5+x3+x1*x4+x2*x4,
(x3+x4*x5+x1*x2*x5)/(x4+x1*x2),
-(x1*x2*x4-x5)*x3/x1/x2,
(x3+x1*x5+x2*x5)/(x1+x2)/x4,
(x4-x1*x2*x5)*x3/x5,
-x4*x5+x1*x2*x5+x3,
x1*x2*x3/(x1*x2*x4-x5),
(x1+x2+x3*x4)/x3/x5,
(-x5+x1*x3-x1*x4+x2*x3-x2*x4)/(x3-x4),
-x3/(x5-x1-x2)/x4,
(x4-x1*x2*x3+x3*x5)/(x5-x1*x2),
-x3*x4/(x3*x5-x1-x2),
(x1*x3+x2*x3+x4*x5)/x4,
(x3-x4*x5-x1*x2*x4)/x4,
x3*x4/(x1*x3+x2*x3-x5),
-x5/(x1*x3+x2*x3-x4),
(x3-x5-x1*x4-x2*x4)/(x1+x2),
(x3*x4+x1*x2*x5)/x3/x5,
-(-x1*x4-x2*x4+x3)*x5/(x1+x2),
-(x3-x4*x5)*x1*x2/x4,
(-x1*x3-x2*x3-x5+x3*x4)/x3,
-(-x1*x2+x3*x4-x3*x5)/(x4-x5),
x5/(x3-x1-x2)/x4,
(x5-x1*x2*x3+x3*x4)/x3,
(-x4+x3+x1*x5+x2*x5)/x5,
x5+x3*x4-x1*x4-x2*x4,
-(x3+x4*x5-x1*x5-x2*x5)/(x4-x1-x2),
x5-x3*x4-x1*x2*x3,
(x3+x1*x2)*(x4-x5),
-(x5+x3*x4-x1*x2)/x3,
x1*x4+x2*x4-x3*x4-x5,
-x4/(x3+x1*x2-x5),
-x4*x5/(x3-x1*x2*x5),
x3-x1*x2*x4+x1*x2*x5,
-(-x4*x5+x1*x2*x5+x3)/(x4-x1*x2),
(x3*x5-x1*x2*x3-x4)/x3,
-(-x1*x4*x5-x2*x4*x5+x3)/(x1+x2)/x5,
(x3-x4)*x5/x1/x2,
(-x1*x4-x2*x4+x3)*x5,
(x1*x3+x2*x3-x4*x5)/x5,
-(x1*x3+x2*x3-x5)*x4/x3,
(x3*x4-x1-x2)/x3/x5,
x1*x2*x5/(x3-x1*x2*x4),
-(x3-x1*x2)/(x4-x5),
x3*x4/(x5+x1*x3+x2*x3),
(x3*x5+x1*x2*x3-x4)/x3,
-(x1*x2*x3-x4*x5)/x3/x4,
(x1*x3+x2*x3-x4)*x5/x3,
x3*x4-x4*x5-x1-x2,
(x4+x1*x5+x2*x5)*x3,
(-x1*x2+x3*x4-x3*x5)/(x4-x5),
(x4+x3*x5-x1-x2)/x5,
-(x3-x1*x2-x4*x5)/x5,
-(x4+x3*x5-x1*x5-x2*x5)/x5,
(x4-x1*x2*x5)/x1/x2/x3,
(x4*x5+x1*x2*x3)/x1/x2/x5,
x1+x2+x3*x4-x3*x5,
-(x1*x3+x2*x3-x4*x5)/x3/x5,
(x5+x1*x2*x4)*x3/x4,
-(x3-x5)/(x4+x1*x2),
(x3-x5-x1*x2)*x4,
(x1*x2+x3*x5)/x4/x5,
(x1*x2*x3-x5)*x4/x3,
(x1*x3+x2*x3+x4*x5)/(x1+x2)/x5,
-(x3+x1*x4-x1*x5+x2*x4-x2*x5)/(x1+x2),
x1*x2*x4-x3*x4-x5,
-(x3-x1*x2*x4)*x5/x1/x2,
x3*x4/(x1+x2+x4*x5),
(-x1*x5-x2*x5+x3*x4)/x4,
(x1+x2)*x5/(x3-x4)
]: