GENERAL PROCEDURE

For example, in the Leech problem on right-angled triangles, we have to find solutions to

                                h² = p ⁴ + (4n² -2)  p²q² + q⁴

which can be clearly written as the general quartic above, and n is a parameter which we choose.

2. Look for simple rational solutions of the quartic. There are often solutions when one variable is 0, eg p = 1, q = 0, h = 1 for the Leech problem, but which lead to non-realistic solutions (length of side of triangle=0).

We could search for non-trivial solutions, but this can take time.

We really need answers to two fundamental questions:
 
(a) do non-trivial solutions exist?
 
(b) if a non-trivial solution exists, how hard is it likely to be to find it?

Elliptic curves provide the means for answering these questions.

3. If we can find one rational solution (even a trivial one) to the above quartic, then we can find a birational transform from the quartic to the elliptic curve

w2 =   v 3 + q v2 + r v + s

  where we assume that we have scaled the variables so that q, r, and s are all integers.

4. Investigate this curve. The theory of elliptic curves is enormous (and can be very intimidating), but we only use a small number of ideas.

The rational points on the curve form a group with

(a) points of finite order, called torsion points ,

(b) points of infinite order, which are generated by r basic points, with r known as the rank of the curve. Note that it is quite common for r = 0, and so no points of infinite order exist.

5. Concentrate initially on the torsion points. Firstly, we should note that, for the form of the curve assumed here, the Lutz-Nagell theorem states that torsion points will have v and w integers. This does NOT mean that all integer points are torsion points, but is does mean that a rational point with non-integer v MUST be of infinite order - a very useful test. It is usually quite simple for a program to find the torsion points.

The torsion points form a subgroup of the rational points group. There is a very famous theorem of Barry Mazur which states that the torsion subgroup is isomorphic to one of the following:
(i) Z n, where n=1,2,3,4,5,6,7,8,9,10,12.
(ii) Z 2 x Z2n with n=1,2,3,4.

Normally, these points of finite order lead to trivial solutions of the underlying equations, so we tend to find realistic solutions to the problem only when the rank of the curve is greater than 0.

6. To estimate the rank, we use the Birch and Swinnerton-Dyer conjecture (see the description ). We produce a list of curves with estimated rank > 0. For each of these curves we also produce an estimate of the "height" of a point of infinite order. This gives an idea of the size of the numbers involved. Basically, the bigger the height, the larger the numbers and the longer it will take us to find the points.

7. For curves with rank greater than 0, we then need to find a rational point of infinite order to give a solution to the original problem.

If rank > 1, this is "usually" fairly easy and can be done mostly by a simple search.

If rank = 1, we apply a series of techniques of increasing complexity. We could have just used John Cremona's wonderful mwrank program, but you learn far more about the fine details of a problem by trying to code it. The methods I use are described in this page .

8. For curves with non-torsion points, use these to find a solution to the original problem. Note that the point of infinite order might NOT give a solution immediately - you might have to add it to one of the torsion points to get a solution.

9.For rank = 1, there are often several curves in each of the problems which still need a rational point of infinite order. Lists of these unsolved curves are included in the discussion of each problem. If anyone finds a solution to one of these curves, please e-mail the author at allan.macleod@paisley.ac.uk.