(* You should be able to copy and paste this text file directly into Mathematica.  These notes will be interpreted as commented code. *)


(* ********* Here are some useful commands for finding points on elliptic curves ******** *)

FindInstance[y^2 == x^3 - 43 x + 166, {x,y}, Integers]

(* This attempts to find an integral solution to y^2 == x^3 - 43 x + 166 *)


FindInstance[y^2 == x^3 - 43 x + 166, {x,y}, Integers, 5]

(* This attempts to find five integral solutions to y^2 == x^3 - 43 x + 166 *)

(* *** Note on FindInstance: Mathematica usually doesn't do a brute-force search, so it will sometimes be unable to find solutions even if there are small ones.  To force it to do an explicit search, you can include a range of values for one of the variables *)

FindInstance[y^2 == x^3 - 43 x + 166 && -100 <= x <= 100, {x,y}, Integers, 5]

(* This attempts to find five integral solutions to y^2 == x^3 - 43 x + 166 with -100 <= x <= 100 *)

(* You can also use Reduce to find solutions when you include a range *)

Reduce[y^2 == x^3 - 43 x + 166 && -100 <= x <= 100, {x,y}, Integers]

(* Reduce will find all of the solutions in the given range *)






(* ***** Here are some useful other commands for problem 1 ****** *)

ellcurvedisc[f_] := -16 * Discriminant[f,x]

ellcurvedisc[x^3 + x^2 - x + 7]

(* Computes the discriminant of the elliptic curve y^2 = f(x) in general Weierstrass form *)





(* ***** Here are some useful other commands for problem 3 ****** *)

poly3a[A_,B_] := 12x(x^3 + A x + B) - (3x^2 + A)^2

poly3c[A_,B_] := 12x(3x^2 + A)(x^3 + A x + B) - (3x^2 + A)^3 - 8(x^3 + A x + B)^2

(* These are the two polynomials in the problem *)




(* ***** Here are some useful other commands for problem 4 ***** *)

(* even n *)
nval = 34;
Reduce[4 x^2 + y^2 + 32 z^2 == nval/2, Integers]
Reduce[4 x^2 + y^2 + 8 z^2 == nval/2, Integers]

(* odd n *)
nval = 13;
Reduce[2 x^2 + y^2 + 32 z^2 == nval, Integers]
Reduce[2 x^2 + y^2 + 8 z^2 == nval, Integers]

(* Computes the solutions to the equations needed for Tunnell's theorem *)

nval = 5;
FindInstance[y^2 == x^3 - nval^2 * x && y != 0, {x,y}, Integers]

(* Searches for an integral point on the congruent-number curve for the given value of n *)
(* As above, you may wish to include a search range for x to force Mathematica to search exhaustively *)