Let the base of the triangle have length b, and let a be the height of the altitude above the base. This is mentioned in section D18 of Richard Guy's book, see [guy].
PROBLEM: What integers n give base/height=n for integer-sided triangles?
The following discussion just provides the basic details of the analysis. Much more detailed accounts are available - PDF version or Postscript version.
Now, we have
|
|
Now, if base/height = n, the second equation is
|
For altitudes outside the triangle the equations are the same, except for z-na replaced by z+na. We thus consider the general system, with n positive or negative.
|
|
Assuming a = 2 p q and z = p2 - q2, then the equation for d is
|
This has an obvious rational point ( p, q, d ) = ( 0, 1, 1 ), so is birationally equivalent to an elliptic curve. Using standard algebra, we can can link this equation to the curve
|
via the transformations p / q = ( n x + y ) / ( x + 1 ).
If, however, a = p2 - q2 and z = 2 p q, we have a different quartic for d2, but leading to the same elliptic curve, with the relevant transformation
p / q = ( n x + x + y + 1 ) / ( n x - x + y - 1 ).
Thus the existence of solutions to the original problem is related to the rational points lying on the curve. There is the obvious point ( x, y ) = ( 0, 0 ), which gives p / q = 0 or p / q = -1, neither of which give non-trivial solutions. A little thought shows the points ( -1, ± n ), giving p / q = ?, p / q = 0 / 0, or p / q = 1, again failing to give non-trivial solutions.
It can be shown that the torsion subgroup must consist
of the point at infinity,
( 0, 0 ), ( -1, ± n
). These points all lead to trivial solutions.
Thus, a non-trivial solution exists iff the rank of En
is at least 1. If the rank is zero then no solution exists.
If we wish (altitude / base) = m, then we can use
the previous theory, with
n = 1 / m.
Defining v = m3 y, u = m2 x, we get the system of elliptic curves Fm, given by
|
For a = 2 p q and z = p2 - q2 , the relevant transformation is
p / q = ( u + v ) / ( m3 + m u ).
p / q = - ( m3 + m u + u + v ) / ( m3 + m u - u - v ).
As an example, for m = 13, the curve is v2 = u3 + 339 u2 + 28561u, which has the point u = 19773/1156, v = 30259281/39304.
The first transformation just given leads to p / q = 741/2278. This gives a = 3375996, so that b = 259692, and x = -4640203. This gives c = 5738365, d = 5950321.
The second transformation leads to the same triangle scaled by a factor of 2.
For both sets of curves, we used the Birch and Swinnerton-Dyer
conjecture to estimate the rank for n and m in the range 2 to 999. The
distribution of the ranks for En and for Fm are given
in the following tables.
| N | No. rank=0 | No. rank=1 | No. rank>1 |
| 2-99 | 43 | 45 | 10 |
| 100-199 | 35 | 51 | 14 |
| 200-299 | 31 | 57 | 12 |
| 300-399 | 44 | 46 | 10 |
| 400-499 | 41 | 52 | 7 |
| 500-599 | 29 | 59 | 12 |
| 600-699 | 40 | 42 | 18 |
| 700-799 | 39 | 50 | 11 |
| 800-899 | 44 | 46 | 10 |
| 900-999 | 44 | 50 | 6 |
| N | No. rank=0 | No. rank=1 | No. rank >1 |
| 2-99 | 48 | 43 | 7 |
| 100-199 | 44 | 51 | 5 |
| 200-299 | 44 | 46 | 10 |
| 300-399 | 45 | 45 | 10 |
| 400-499 | 34 | 55 | 11 |
| 500-599 | 47 | 46 | 7 |
| 600-699 | 44 | 51 | 5 |
| 700-799 | 52 | 39 | 9 |
| 800-899 | 47 | 46 | 7 |
| 900-999 | 48 | 46 | 6 |
The solutions to the base/altitude problem can be found here, whilst the solutions for the altitude/base problem are here.
There are many values of N which have curves with rank 1 but no known generator. For the En curves, there are only 53 unsolved values of n ( as of May 2006), all with predicted heights over 110, with the maximum height estimated as 630. I have been really lucky with these curves as practically all problem values had a 2-isogenous curve with a point half the height, which was reasonably easy to find. As of August 2010, the number of unknowns is down to 13!
For Fm , however, the problem is significantly worse, with , currently, some 209 values of m unsolved. The 2-isogenous curve for these problems gives (usually) a height estimate of double the original curve. It is also the case that the heights of rank 1 generators are much larger for this family of curves. For example, the Birch and Swinnerton-Dyer conjecture predicts a height of 3580 for m = 919. I am gradually reducing these, but it takes time.