Skip to main content
1 of 5
Will Jagy
  • 25.7k
  • 2
  • 65
  • 121

Spent a month checking, this is what I suspect is the complete list of 'sporadic' or 'exceptional' pairs.

---------------------------------------------------------------
111 :   1   4     7   1  0  0       75 :  1   4     5   1  1  0
142 :   3   3     5   2  3  1       78 :  3   3     3   1  1  3
158 :   3   3     5  -1  2  1       78 :  3   3     3   1  1  3
158 :   3   3     5  -1  2  1      142 :  3   3     5   2  3  1
190 :   3   5     5   5  2  3       78 :  3   3     3   1  1  3
190 :   3   5     5   5  2  3      142 :  3   3     5   2  3  1
190 :   3   5     5   5  2  3      158 :  3   3     5  -1  2  1
213 :   2   4     7   0  1  1      177 :  2   4     7   4  2  1
216 :   2   4     8   4  1  1       54 :  2   2     4   1  2  0
232 :   3   5     5   3  1  3      232 :  3   3     7   1  2  1
284 :   3   5     6   4  2  2      156 :  3   3     5   2  2  0
316 :   3   5     6   0  2  2      156 :  3   3     5   2  2  0
316 :   3   5     6   0  2  2      284 :  3   5     6   4  2  2
333 :   3   4     7   1  0  0      225 :  3   4     7   4  3  3
380 :   3   5     7   2  0  2      156 :  3   3     5   2  2  0
380 :   3   5     7   2  0  2      284 :  3   5     6   4  2  2
380 :   3   5     7   2  0  2      316 :  3   5     6   0  2  2
567 :   4   6     7   3  2  3      324 :  4   4     6   0  3  2
639 :   5   5     8  -1  2  4      531 :  5   5     6   0  3  2
648 :   2   6    14   3  1  0      162 :  2   2    14   1  2  2
648 :   5   7     7   6  1  5      648 :  5   5     8   0  4  3
999 :   5   8     8  -5  1  4      675 :  5   5     8  -1  4  2
1944 :  2   6    41   3  1  0      486 :  2   2    41   1  2  2
2592 :  4   7    25  -4  2  2      648 :  4   7     7   5  2  2
These are pairs of positive quadratic forms that represent the
same numbers, and violate a Kaplansky conjecture.  

Delta : A B C R S T     means

f(x,y,z) = A x^2 + B y^2 + C z^2 + R y z + S z x + T x y,

and Delta = 4ABC + RST - A R^2 - B S^2 - C T^2.

The two pair within a genus each are
232 :   3   5     5   3  1  3      232 :  3   3     7   1  2  1
648 :   5   7     7   6  1  5      648 :  5   5     8   0  4  3

The most productive discriminant ratio  is 4, 
which includes Kap's two infinite families, also
 24 :   1   2     4   2  1  1        6 :  1   1     2   1  1  0  
 72 :   2   2     5   1  1  1       18 :  2   2     2   1  2  2
216 :   2   5     6   3  0  1       54 :  2   2     5   1  2  2 
648 :   2   6    14   3  1  0      162 :  2   2    14   1  2  2
1944 :  2   6    41   3  1  0      486 :  2   2    41   1  2  2
or
48N-24: 2   6    N    3  1  0     12N-6:  2   2     N   1  2  2  
where N = (1+ 3^k)/2, and the pairs for N = 1,2,5 are regular 
and have been Schiemann reduced.
------------------------------------------------------------
Reminder: Kap's two infinite families are equivalent to those 
below, which need not be "reduced." For the first, require
gcd(A,C) = 1 and 0 <A,C. For the second, gcd(A,R) = 1, with
A > 0 and  -A < R < 2 A.
  
4D : A    3A    C   0   0  0    D :   A   A   C    0    0    A

4D:  A  2A-R  2A+R  0  2R  0    D :   A   A   A    R    R    R

For the first, D = 3 A^2 C, for the second D = (A+R)(2A-R)^2 .
Will Jagy
  • 25.7k
  • 2
  • 65
  • 121