x2 + N y2   and  x2 + N2 y2  both  squares



The problem of the title is one which seems a reasonable one to ask, but I can find no reference to it even in Dickson's History. This does not mean that the problem has never been considered - I just can't find any mention.

Consider first  x2 + N2y 2 = b2 , and define  w =  b/x and  z = y/x , so that  w2 = N2z2 + 1, which has the obvious rational solution z = 0 , w =  1.

The line  w = 1 + m z  meets  w2 = N2 z2 + 1  at  z = 0  and  z = 2 m / ( N2 - m2 ).

Define m = p / q,  then  z = y / x = 2 p q / ( N2 q2 - p2 ),  giving a parameterisation of
x2 + N2 y2 = b2   as  x = N2 q2 - p2  and  y = 2 p q.

Substituting these into  x2 + N y 2 = a2, we find

                                   a2  =  p4  +  2 N ( 2 - N )  p2 q2  +  N4 q4
Define  k = a / q2 ,  and use  m = p / q  giving

                                            k2  =  m4  +  2 N ( 2 - N ) m2  +  N4
This quartic has the obvious rational solution  m = 0 ,  k = N2.

Thus, the quartic is birationally equivalent to the elliptic curve

           EN : g2  =  h3  +  N ( N - 2 ) h2  +  N2 ( 1 - N ) h 
also written as

                         EN : g2  =  h ( h - N ) ( h + N (N-1) )
with the transformation m=p/q=g/h.

The three rational roots imply at least 4 torsion points, namely the pt. at infinity,  ( 0 , 0 ) , ( N , 0 ), and ( -N ( N - 1 ) , 0 ).

Numerical evidence suggests these are the only ones, but I am unable to prove this. If this is so, then to get a non-trivial solution we need to have EN to have rank at least 1.

For example, the Birch and Swinnerton-Dyer conjecture shows that EN has rank at least 1 for N=74.

We find the following point on the curve  h = 13686697 / 11664  ,  g = 116021624725 / 1259712.

This gives  p = 3670525 and  q = 46764 , and hence  x = 1497444368329 ,  y = 343296862200, a = 3311108888329  and  b = 25448063182921.

We used the BSD conjecture to estimate the rank for  2  £  N  £  999 . The distribution of ranks is given in the following table.
 
 


      N range    Rank = 0   Rank = 1  Rank > 1
    [ 2 , 99 ]        44         49           5
[ 100 , 199 ]        39         51         10
[ 200 , 299 ]        36         56           8
[ 300 , 399 ]        37         48         15
[ 400 , 499 ]        39         49         12
[ 500 , 599 ]        43         47         10
[ 600 , 699 ]        40         52           8
[ 700 , 799 ]        39         50         11
[ 800 , 899 ]        36         57           7
[ 900 , 999 ]        45         45         10

The solutions found for the values of N with rank greater than 0 can be found in the following file .

There are several values of N where the Birch and Swinnerton-Dyer conjecture predicts a rank greater than 0 but where we cannot find a solution. Currently, there are 73 values of N unsolved. Many of these curves have predicted heights of several hundred which means we are unlikely to be able to find a solution, so we only give the smallest 10 heights (updated 26 August 2004).
 
 
 


        N     Height                   Information 
      759       30  
      938       37  
      795       44  
      602       48  
      706       48  
      844       48  
      327       50  
      397       52  
      509       52  
      515       53  

.Latest revision:  26 August , 2004