Base/height or height/base = n

1  Introduction

Consider the following triangle.

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
                                                a2  +  z2  =  c2
                                         a2  +  ( b - z )2  =  d2

Now, if base/height = n, the second equation is
                                       a2  +  ( z - n a )2  =  d2

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.
                                          a2  +  z2  =  c2
                                   a2  +  ( z - n a )2  =  d2
Clearly, we can assume that a and z have no common factors, so there exists integers p and q (of opposite parities) such that
(1) a = 2 p q  ,  z = p2 - q2,
(2) a = p2 - q2 ,  z = 2 p q.

Assuming a = 2 p q and z = p2 - q2, then the equation for d is

                     d2  =  p4 - 4 n p3 q  +  ( 4 n2 + 2 ) p2 q2 +  4 n p q3  +  q4

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
                En  :  y2  =  x3  +  ( n2 + 2 ) x2  +  x

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
                  Fm : v2  =  u3 +  ( 2 m2 + 1 ) u2  +  m4 u

For  a = 2 p q and z = p2 - q2 , the relevant transformation is


p / q = ( u + v ) / ( m3 + m u ).

For  a = p2 - q2 and z = 2 p q , the transformation is

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.

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.

File translated from TEX by TTH, version 2.86.
Latest revision: 8 May  2006.