#C22.txt, April 14, 2025
Help22:=proc(): print(` BAc(f), Likelyc(q,r) , LikelycC(q,r) , ConfirmR(q,r). ExperimentS4() `): 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:


#From Joseph K
#5

RM := proc() local ra:
  ra := rand(1..10):
  [[ra(),ra()],[ra(),ra()]]:
end:

RMs := proc(n) local i:
  [seq(RM(), i=1..n)]:
end:

#ALn(X,n): rediscovering the Amitsur-Levitsky identity for n by n matrix
ALn := proc(X,n) local A, pi, eqs, M, var, i, var1, v, T,JK:
  var := {seq(c[pi], pi in permute(2*n))}:
JK:=add(c[pi]*X[pi],pi in permute(2*n)):

  eqs := {}:

  for i from 1 to trunc((2*n)!/n^2)+1 do:
    A := RMs(2*n):
    M := expand(add(c[pi]*Mul(Mul(A[pi[1]],A[pi[2]]),Mul(A[pi[3]],A[pi[4]])), pi in permute(4))):
    eqs := eqs union {op(M[1]), op(M[2])}:
  end:
  var := solve(eqs, var):

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

  for v in var do:
    T[op(1, v)] := subs({var1[1]=1,seq(var1[i]=0,i=2..nops(var1))},op(2,v)):
  od:
  subs(op(T),JK):
end: