#C21.txt, April 10, 2025
Help21:=proc(): print(`Eq510(q,n,c,x), Eq510(256,33,10,5) , Histogram510(q,n,x), BA1(f,r) , BA(f) , CF(f) `):
CVG(f):
end:
read `C17.txt`:
Digits:=20:

Eq510:=proc(q,n,c,x) local r,u:
r:=Ord(x,n):
evalf(abs(1/r*int(exp(2*Pi*I*mods(r*c,q)/r*u),u=0..1))^2):
end:

Histogram510:=proc(q,n,x) local c:
plot([seq([c,Eq510(q,n,c,x)],c=1..q-1)]);
end:

#BA1(f,r): the closest fraction d/r to f, where r is fixed
BA1:=proc(f,r) local b: 
b:=round(f*r)/r: 
[b, evalf(abs(b-f)*denom(f))]
end:

#BA(f): the best fraction d/r where r is between 1 and sqrt(denom(f))
BA:=proc(f) local r,cha,rec,hope:
cha:=BA1(f,1)[1]:
rec:=BA1(f,1)[2]:
for r from 2 to trunc(sqrt(denom(f))) do
 hope:=BA1(f,r):
 if hope[2]<rec then
   cha:=hope[1]:
   rec:=hope[2]:
 fi:
od:
cha,rec:
end:

#CF(f): inputs an integer or fraction outputs the list consisting of the continued-fraction
#11/4=2+3/4= 2+ 1/(4/3)= 2+ 1/(1+1/3) f=a1+1/(a2+ 1/(a3+ 1/....)
CF:=proc(f) local a:
if not (type(f,rational) and f>0) then
 RETURN(FAIL):
fi:
if type(f,integer) then
   RETURN([f]):
else
a:=trunc(f):
RETURN([a,op(CF(1/(f-a)))]):
fi:
end:

#EvalCF(L): inputs a list of pos. integers and outputs the fraction whose cont. fraction it is
#(reverse of CF(f))
EvalCF:=proc(L)
if nops(L)=1 then
 L[1]:
else
L[1]+1/EvalCF([op(2..nops(L),L)]):
fi:
end:

#CVG(f): inputs a rational number f and outputs its list of CONVERGENTS
CVG:=proc(f) local L,i:
L:=CF(f):
[seq(EvalCF([op(1..i,L)]),i=1..nops(L))]:
end:

#From C20.txt 
#C20.txt, April 7, 2025
Help20:=proc(): print(`Eq55(q,n,k,c,x), Eq56(q,n,k,c,x) , Eq57(q,n,k,c,x), Eq58(q,n,k,c,x), Eq59(q,n,k,c,x), Histogram56(q,k,n,x)  `):end:
read `C17.txt`:

#Eq55(q,k,n,x): the value of Eq. (5.5) in Shor's paper
Eq55:=proc(q,n,k,c,x) local a,su:
su:=0:
for a from 0 to q-1 do
 if x^a-x^k mod n=0 then
   su:=evalc(su+exp(2*Pi*I*a*c/q)):
 fi:
od:
evalf(abs(su/q)^2):
end:



#Eq56(q,k,n,c,x): the value of Eq. (5.5) in Shor's paper
Eq56:=proc(q,n,k,c,x) local r,b:
r:=Ord(x,n):
if k>=r then
  RETURN(FAIL):
fi:

evalf(abs(add(exp(2*Pi*I*(b*r+k)*c/q),b=0..trunc((q-k-1)/r))/q)^2):
end:

#Eq56(q,k,n,c,x): the value of Eq. (5.6) in Shor's paper
Eq56:=proc(q,n,k,c,x) local r,b:
r:=Ord(x,n):
if k>=r then
  RETURN(FAIL):
fi:

evalf(abs(add(exp(2*Pi*I*(b*r+k)*c/q),b=0..trunc((q-k-1)/r))/q)^2):
end:

#Eq57(q,k,n,c,x): the value of Eq. (5.7) in Shor's paper
Eq57:=proc(q,n,k,c,x) local r,b:
r:=Ord(x,n):
if k>=r then
  RETURN(FAIL):
fi:

#evalf(abs(add(exp(2*Pi*I*(b*({r*c}_q)/q),b=0..trunc((q-k-1)/r))/q)^2):
evalf(abs(add(exp(2*Pi*I*b*(mods(r*c,q))/q),b=0..trunc((q-k-1)/r))/q)^2):
end:

#Eq58(q,k,n,c,x): the value of Eq. (5.8) in Shor's paper (w/o the error term)
Eq58:=proc(q,n,k,c,x) local r,b:
r:=Ord(x,n):
if k>=r then
  RETURN(FAIL):
fi:

evalf(abs(int(exp(2*Pi*I*b*(mods(r*c,q))/q),b=0..trunc((q-k-1)/r))/q)^2):
end:

#Eq59(q,k,n,c,x): the value of Eq. (5.9) in Shor's paper (w/o the error term)
Eq59:=proc(q,n,k,c,x) local r,b,u:
r:=Ord(x,n):
if k>=r then
  RETURN(FAIL):
fi:

evalf(abs(int(exp(2*Pi*I*(mods(r*c,q))/r),u=0..r/q*trunc((q-k-1)/r))/q)^2):
end:


#Added after class
#Histogram56(q,k,n,c): the plot of [c,Eq56(q,k,n,c,x)] for c from 1 to q-1
Histogram56:=proc(q,k,n,x) local c:
plot([seq([c,Eq56(q,k,n,c,x)],c=1..q-1)]);
end:

#end C20.txt