# Please do not post homework
# Aurora Hiveley, 04/01/24, Assignment 20

Help:= proc(): print(` farFromC(A,B,C), IntSol(a,b,K), SolChain(a,b,S1,S2,K)  `): end:

### plot figure 3 activity

# helper function which returns which of A or B is further from C on the plot 
# will be used to draw the line through A-B-C
farFromC := proc(A,B,C) :
if evalf(abs(A[1] - C[1])) < evalf(abs(B[1] - C[1])) then
    B:
elif evalf(abs(A[1] - C[1])) > evalf(abs(B[1] - C[1])) then
    A:
else
    RETURN(FAIL):
fi:
end:

### build plot
# with(plots):
# with(plottools):
## create elliptic curve
# a := 10:
# b := 50:
# f := implicitplot(y^2 = x^3 + a*x + b): 
## pick two points A,B and build A+B and C accordingly
# A := RP(a,b):
# B := RP(a,b):
# AplusB := AddE(a,b,A,B):
# C := [AplusB[1], -AplusB[2]]:
## text labels for each point
# tA := textplot([op(A + [0,0.25]),'A'],'align'={'above'}, 'font' = ['times', 'new', 16]):
# tB := textplot([op(B + [0,0.25]),'B'],'align'={'above'}, 'font' = ['times', 'new', 16]):
# tAB := textplot([op(AplusB + [0,-0.25]),'A+B'],'align'={'below'}, 'font' = ['times', 'new', 16]):
# tC := textplot([op(C + [0,0.25]),'C'],'align'={'above'}, 'font' = ['times', 'new', 16]):
## lines through A-B-C and from A+B to C
# l1 := line(AplusB,C,color="Blue"):
# l2 := line(C,farFromC(A,B,C), color="Green"):

## display all of the above on one plot
# display({f,tA,tB,tC,tAB,l1,l2,pointplot([A,B,C,AplusB])}, 'view' = [-5..5,-15..15]);




### integer solutions to diophantine activity
# finds all integer solutions to y^2 = x^3 + a*x + b
# between -K and K given input a and b
IntSol := proc(a,b,K) local x,y,sols:
sols := {}:
for x from -K to K do
    y := sqrt(x^3 + a*x + b):

    if type(y,integer) then
        sols := sols union {[x,y]}:
    fi:
od:
sols:
end:

# testing:
# IntSol(1,2,5);        # = {[-1,0], [1,2]}:
# IntSol(10,50,100);    # = {[5,15]}:
# IntSol(3,5,100);      # = {[-1,1], [1,3], [4,9], [11,37]}




### solchain activity
# takes two solutions S1 and S2 of y^2 = x^3 + a*x + b 
# repeatedly uses AddE(a,b,L[-2],L[-1]) to get a list
# of K+2 total solutions in the rationals
SolChain := proc(a,b,S1,S2,K) local sols:
sols := [S1,S2]:

while nops(sols) < K+2 do
    sols := [op(sols), AddE(a,b,sols[-2],sols[-1])]:
od:

sols:
end:

## test calls
# SolChain(3,5,RP(3,5),RP(3,5),6);
# SolChain(2,4,RP(2,4),RP(2,4),3);

# there are definitely some solutions which do not produce a third
# for example, the integer solutions from above caused errors in AddE
# but this doesn't gel with the idea that any line through the curve 
# intersects at three points. if this claim is true, then we could
# go on infinitely. the integer error is probably because at some point 
# you get two points which are reflections of each other over the x-axis
# i.e. P[2] = -Q[2], so there is no third point of intersection.
# obviously we remedied this in class on thursday.



### COPIED FROM CLASS

Help20:=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: