#Dayoon Kim, 2024-4-7, HW20
Help:=proc(): print(` Diag(a,b),IntSol(a,b,K),SolChain(a,b,S1,S2,K) `): end:

#####Old Code
#C20.txt, April 1, 2024;
Help20:=proc(): print(`RP(a,b), DoesLie(a,b,P), AddE(a,b,P,Q)`): end:

#random number
RP:=proc(a,b) local x: x:=rand(0..200)()/100: [x,sqrt(x^3+a*x+b)]: end:

#checking if the point lies in the curve
DoesLie:=proc(a,b,P): P[2]^2-P[1]^3-a*P[1]-b: end:

#
AddE:=proc(a,b,P,Q) local y,s,sol,xR:
 s:=(P[2]-Q[2])/(P[1]-Q[1]):
 y:=P[2]+(x-P[1])*s:
 sol:={solve(y^2-x^3-a*x-b,x)}:
 #sol: finding 3 x coordinate of points where elliptic curve and the line meets
if not (nops(sol)=3 and member(P[1],sol) and member(Q[1],sol)) then
  print(`sth bad happened`):
  RETURN(Fail):
fi:

sol:=sol minus {P[1],Q[1]}:
xR:=sol[1]:
[xR,sqrt(xR^3+a*xR+b)]:
end:
#####End of Old Code

#HW20-1
Diag:=proc(a,b) local A,B,L,f:
A:=RP(a,b):
B:=[A[1],-A[2]]:
L:=[A,B]:
f:=sqrt(x^3 + a*x + b):
print(`Random point=`, A);
plot([f,-f,L]);
end:

#HW20-2
IntSol:=proc(a,b,K) local sol,x,y:
 sol:=[]:
 for x from -K to K do
   y:=floor(sqrt(x^3 + a*x + b)):  # Integer square root
   if y^2=x^3+a*x+b then
     sol:=[op(sol),[x,y]]:
   fi:
 od:
sol;
end:

#HW20-3
SolChain:=proc(a,b,S1,S2,K) local L, i, P, Q:
  L:=[S1, S2]:
  for i from 3 to K + 2 do
    P:=L[-2]:
    Q:=L[-1]:
    L:=[op(L), AddE(a, b, P, Q)]:
  od:
L;
end:

#No, you cannot get infinitely many solutions.
#IntSol(2, 3, 15);
#[[-6, 15 I], [-2, 3 I], [-1, 0], [3, 6]]
#SolChain(2,3,[-1, 0],[3, 6], 9);
#[[-1, 0], [3, 6], [1/4, 15/8], [-1, 0], [3, 6], [1/4, 15/8], [-1, 0], [3, 6], [1/4, 15/8], [-1, 0], [3, 6]]
#Same three repeats.