#C20.txt: April 1, 2024
Help:=proc(): print(`RP(a,b), DoesLie(a,b,P), AddE(a,b,P,Q) `):end:
#Elliptic curve: y^2=x^3+a*x+b 
RP:=proc(a,b) local x: x:=rand(0..200)()/100: [x,sqrt(x^3+a*x+b)]:end:

DoesLie:=proc(a,b,P): P[2]^2-P[1]^3-a*P[1]-b:end:

AddE:=proc(a,b,P,Q) local x,y,s,eq,sol,R,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)};


if not (nops(sol)=3 and member(P[1],sol) and member(Q[1],sol)) then
  print(`Something bad happened`):
  RETURN(FAIL):
fi:

sol:=sol minus {P[1],Q[1]}:
xR:=sol[1]:
[xR,-sqrt(xR^3+a*xR+b)]:

end: