Spent a month checking, this is what I suspect is the complete list of 'sporadic' or 'exceptional' pairs. No restriction that they be in the same genus or have the same discriminant. I was able to check discriminant ratio $4$ and discriminant ratio $1$ very high. The other ones just seem sort of random little sets, two quadruples (see the repeated forms). Also some patterns that dry up, discriminants $111,333,999$ but not $2997,$ also on the left $24, 72, 216, 648, 1944,$ but not $5832.$ This last begins with three pair of regular forms, I typed those in. The list of pairs of regular forms that agree is enormous.
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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.
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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 .
why not. The complete list of regular pairs, evidently 182 pairs among which various triples, quadruples and so on may be found. Smaller discriminant put first on the line, which is why i had to delete the colons :, to use Unix sort -n
2 1 1 1 1 1 1 18 1 2 3 2 1 0
2 1 1 1 1 1 1 32 1 2 4 0 0 0
2 1 1 1 1 1 1 8 1 1 2 0 0 0
3 1 1 1 0 0 1 12 1 1 3 0 0 0
3 1 1 1 0 0 1 12 1 2 2 1 1 1
3 1 1 1 0 0 1 21 1 2 3 0 0 1
4 1 1 1 0 0 0 16 1 2 2 0 0 0
4 1 1 1 0 0 0 36 1 2 5 2 0 0
5 1 1 2 1 1 1 20 1 2 3 1 0 1
5 1 1 2 1 1 1 20 1 2 3 2 0 0
6 1 1 2 0 0 1 22 1 2 3 0 1 0
6 1 1 2 0 0 1 24 1 2 3 0 0 0
6 1 1 2 1 1 0 15 1 1 4 0 1 0
6 1 1 2 1 1 0 24 1 1 6 0 0 0
6 1 1 2 1 1 0 24 1 2 4 2 1 1
6 1 1 2 1 1 0 33 1 2 5 1 1 1
8 1 1 2 0 0 0 18 1 2 3 2 1 0
8 1 1 2 0 0 0 32 1 2 4 0 0 0
8 1 1 3 1 1 1 32 1 3 3 2 0 0
8 1 1 3 1 1 1 72 1 3 7 2 1 1
9 1 1 3 0 0 1 36 1 3 3 0 0 0
9 1 1 3 0 0 1 36 1 3 4 3 1 0
9 1 1 3 0 0 1 63 1 3 6 3 0 0
10 1 2 2 2 1 1 15 1 2 2 1 0 0
10 1 2 2 2 1 1 40 1 2 5 0 0 0
12 1 1 3 0 0 0 12 1 2 2 1 1 1
12 1 1 3 0 0 0 21 1 2 3 0 0 1
12 1 1 4 0 0 1 48 1 3 4 0 0 0
12 1 2 2 1 1 1 21 1 2 3 0 0 1
12 1 2 2 2 0 0 44 1 2 6 2 0 0
12 1 2 2 2 0 0 48 1 2 6 0 0 0
14 1 1 5 1 1 1 30 1 3 3 1 1 1
14 1 1 5 1 1 1 46 1 3 5 3 1 1
14 1 1 5 1 1 1 56 1 3 5 2 0 0
15 1 1 4 0 1 0 24 1 1 6 0 0 0
15 1 1 4 0 1 0 24 1 2 4 2 1 1
15 1 1 4 0 1 0 33 1 2 5 1 1 1
15 1 2 2 1 0 0 40 1 2 5 0 0 0
16 1 2 2 0 0 0 36 1 2 5 2 0 0
18 1 1 6 0 0 1 72 1 3 6 0 0 0
18 1 2 3 2 1 0 32 1 2 4 0 0 0
18 2 2 2 1 2 2 45 2 2 3 0 0 1
18 2 2 2 1 2 2 72 2 2 5 1 1 1
18 2 2 2 1 2 2 72 2 3 3 0 0 0
18 2 2 2 1 2 2 99 2 3 5 3 1 0
20 1 1 7 1 1 1 80 1 3 7 2 0 0
20 1 2 3 1 0 1 20 1 2 3 2 0 0
22 1 2 3 0 1 0 24 1 2 3 0 0 0
24 1 1 6 0 0 0 24 1 2 4 2 1 1
24 1 1 6 0 0 0 33 1 2 5 1 1 1
24 1 2 4 2 1 1 33 1 2 5 1 1 1
25 2 2 2 -1 1 1 100 2 2 7 -1 1 1
25 2 2 2 -1 1 1 100 2 3 5 0 0 2
27 1 1 7 0 1 0 27 1 2 4 1 0 1
27 1 1 9 0 0 1 108 1 3 10 3 1 0
27 1 1 9 0 0 1 108 1 3 9 0 0 0
27 1 1 9 0 0 1 27 1 3 3 3 0 0
27 1 3 3 3 0 0 108 1 3 10 3 1 0
27 1 3 3 3 0 0 108 1 3 9 0 0 0
27 2 2 2 1 1 1 108 2 2 8 2 2 1
27 2 2 2 1 1 1 108 2 3 5 0 2 0
27 2 2 2 1 1 1 189 2 3 8 0 1 0
28 2 2 3 2 2 2 112 2 3 5 2 0 0
28 2 2 3 2 2 2 60 2 3 3 0 0 2
28 2 2 3 2 2 2 92 2 3 5 2 0 2
30 1 1 10 0 0 1 120 1 3 10 0 0 0
30 1 3 3 1 1 1 46 1 3 5 3 1 1
30 1 3 3 1 1 1 56 1 3 5 2 0 0
32 1 3 3 2 0 0 72 1 3 7 2 1 1
36 1 1 12 0 0 1 144 1 3 12 0 0 0
36 1 3 3 0 0 0 36 1 3 4 3 1 0
36 1 3 3 0 0 0 63 1 3 6 3 0 0
36 1 3 4 3 1 0 63 1 3 6 3 0 0
36 2 2 3 0 0 2 144 2 3 6 0 0 0
44 1 2 6 2 0 0 48 1 2 6 0 0 0
45 2 2 3 0 0 1 72 2 2 5 1 1 1
45 2 2 3 0 0 1 72 2 3 3 0 0 0
45 2 2 3 0 0 1 99 2 3 5 3 1 0
46 1 3 5 3 1 1 56 1 3 5 2 0 0
48 1 4 4 4 0 0 192 1 4 12 0 0 0
50 1 4 4 3 1 1 200 1 5 10 0 0 0
50 1 4 4 3 1 1 75 1 4 5 0 0 1
54 1 1 18 0 0 1 216 1 3 18 0 0 0
54 1 4 4 2 1 1 216 1 6 9 0 0 0
54 1 4 4 2 1 1 297 1 6 13 3 1 0
54 2 2 5 1 2 2 135 2 5 5 5 1 2
54 2 2 5 1 2 2 216 2 5 6 0 0 2
54 2 2 5 1 2 2 216 2 5 6 3 0 1
54 2 2 5 1 2 2 297 2 5 8 -2 1 1
54 2 3 3 3 0 0 216 2 3 9 0 0 0
60 1 4 5 4 1 0 132 1 5 7 1 0 1
60 1 4 5 4 1 0 96 1 4 7 4 0 0
60 2 2 5 0 0 2 240 2 5 6 0 0 0
60 2 3 3 0 0 2 112 2 3 5 2 0 0
60 2 3 3 0 0 2 92 2 3 5 2 0 2
64 3 3 3 -2 2 2 256 3 3 8 0 0 2
64 3 3 3 -2 2 2 576 3 3 19 -2 2 2
72 2 2 5 1 1 1 72 2 3 3 0 0 0
72 2 2 5 1 1 1 99 2 3 5 3 1 0
72 2 3 3 0 0 0 99 2 3 5 3 1 0
75 1 4 5 0 0 1 200 1 5 10 0 0 0
80 3 3 3 2 2 2 320 3 4 7 0 2 0
90 1 1 30 0 0 1 360 1 3 30 0 0 0
92 2 3 5 2 0 2 112 2 3 5 2 0 0
96 1 4 7 4 0 0 132 1 5 7 1 0 1
98 3 3 3 -1 1 1 392 3 5 7 0 0 2
100 2 2 7 -1 1 1 100 2 3 5 0 0 2
100 3 3 3 1 1 1 400 3 5 7 0 2 0
108 1 1 36 0 0 1 432 1 3 36 0 0 0
108 1 3 9 0 0 0 108 1 3 10 3 1 0
108 1 4 7 0 1 0 108 1 5 7 5 1 1
108 1 6 6 6 0 0 432 1 6 18 0 0 0
108 2 2 8 2 2 1 108 2 3 5 0 2 0
108 2 2 8 2 2 1 189 2 3 8 0 1 0
108 2 2 9 0 0 2 432 2 6 9 0 0 0
108 2 3 5 0 2 0 189 2 3 8 0 1 0
108 3 3 5 3 3 3 432 3 5 8 4 0 0
135 2 5 5 5 1 2 216 2 5 6 0 0 2
135 2 5 5 5 1 2 216 2 5 6 3 0 1
135 2 5 5 5 1 2 297 2 5 8 -2 1 1
144 3 4 4 4 0 0 576 3 4 12 0 0 0
150 2 5 5 5 0 0 600 2 5 15 0 0 0
180 2 2 15 0 0 2 720 2 6 15 0 0 0
180 3 5 5 5 3 3 288 3 5 5 2 0 0
180 3 5 5 5 3 3 396 3 5 8 2 0 3
192 1 8 8 8 0 0 704 1 8 24 8 0 0
192 1 8 8 8 0 0 768 1 8 24 0 0 0
196 3 5 5 -4 2 2 784 3 5 14 0 0 2
216 1 6 9 0 0 0 297 1 6 13 3 1 0
216 2 5 6 0 0 2 216 2 5 6 3 0 1
216 2 5 6 0 0 2 297 2 5 8 -2 1 1
216 2 5 6 3 0 1 297 2 5 8 -2 1 1
240 1 4 16 4 0 0 384 1 4 24 0 0 0
256 3 3 8 0 0 2 576 3 3 19 -2 2 2
270 3 3 11 3 3 3 1080 3 9 11 6 0 0
288 3 5 5 2 0 0 396 3 5 8 2 0 3
300 1 10 10 10 0 0 1200 1 10 30 0 0 0
384 4 4 7 0 4 0 528 4 7 7 6 0 4
400 3 7 7 -6 2 2 1600 3 7 20 0 0 2
432 1 12 12 12 0 0 1728 1 12 36 0 0 0
432 5 5 5 -2 2 2 1728 5 8 12 0 0 4
448 5 5 5 2 2 2 1472 5 8 12 8 4 0
448 5 5 5 2 2 2 1792 5 8 12 0 4 0
448 5 5 5 2 2 2 960 5 5 12 -4 4 2
450 5 5 6 0 0 5 1800 5 6 15 0 0 0
540 5 5 8 2 4 5 1188 5 8 9 6 3 2
540 5 5 8 2 4 5 864 5 8 8 8 2 4
540 6 6 7 6 6 6 2160 6 7 13 2 0 0
576 3 8 8 8 0 0 2304 3 8 24 0 0 0
704 1 8 24 8 0 0 768 1 8 24 0 0 0
720 3 8 8 4 0 0 1152 3 8 12 0 0 0
768 1 16 16 16 0 0 3072 1 16 48 0 0 0
864 5 8 8 8 2 4 1188 5 8 9 6 3 2
900 3 10 10 10 0 0 3600 3 10 30 0 0 0
960 5 5 12 -4 4 2 1472 5 8 12 8 4 0
960 5 5 12 -4 4 2 1792 5 8 12 0 4 0
960 5 8 8 8 0 0 3840 5 8 24 0 0 0
1024 3 11 11 -10 2 2 4096 3 11 32 0 0 2
1152 5 5 12 0 0 2 1584 5 5 17 -2 2 2
1280 7 7 7 -2 2 2 5120 7 12 16 0 0 4
1350 7 7 7 -1 1 1 5400 7 13 15 0 0 2
1472 5 8 12 8 4 0 1792 5 8 12 0 4 0
1728 1 24 24 24 0 0 6912 1 24 72 0 0 0
1728 8 8 9 0 0 8 6912 8 9 24 0 0 0
2160 5 8 17 8 2 4 3456 5 8 24 0 0 4
2304 3 16 16 16 0 0 9216 3 16 48 0 0 0
2700 9 11 11 -8 6 6 10800 9 11 30 0 0 6
2880 8 8 15 0 0 8 11520 8 15 24 0 0 0
3136 3 19 19 -18 2 2 12544 3 19 56 0 0 2
3456 8 11 11 -2 4 4 4752 8 11 15 6 0 4
4800 1 40 40 40 0 0 19200 1 40 120 0 0 0
6144 11 11 16 8 8 6 8448 11 11 19 2 2 6
6144 7 15 16 0 0 6 8448 7 15 23 -6 2 6
6400 3 27 27 -26 2 2 25600 3 27 80 0 0 2
6912 1 48 48 48 0 0 27648 1 48 144 0 0 0
6912 9 17 17 -14 6 6 27648 9 17 48 0 0 6
8640 13 13 13 2 2 2 34560 13 24 28 0 4 0
14400 3 40 40 40 0 0 57600 3 40 120 0 0 0
18432 17 17 20 -4 4 14 25344 17 20 20 -8 4 4
18432 5 20 48 0 0 4 25344 5 20 68 -8 4 4
43200 9 41 41 -38 6 6 172800 9 41 120 0 0 6
55296 11 32 44 -16 4 8 76032 11 32 59 8 10 8