There are three genera of positive, integral, ternary quadratic forms in which both forms (classes...) are regular, so the paired forms represent the same numbers. These pairs (complete genera) are:
$$ x^2 + y^2 + 9 z^2 + xy, \; \; \; x^2 + 3 y^2 + 3 z^2 + 3 y z, $$
$$ x^2 + y^2 + 7 z^2 + zx, \; \; \; x^2 + 2 y^2 + 4 z^2 + y z + xy, $$
$$ x^2 + 4y^2 + 7 z^2 + zx, \; \; \; x^2 + 5 y^2 + 7 z^2 + 5 y z +zx + xy. $$
In these three cases the relationship of represented numbers is proven.
In response to a question by a master's student in New Zealand, originating with his adviser, Steven Galbraith, I have found, so far, two pairs of irregular forms, in the same genus, that represent the same numbers up to $10^6;$ these are
$$ 3x^2 + 3y^2 + 7 z^2 + yz +2zx +xy, \; \; \; 3x^2 + 5 y^2 + 5 z^2 + 3 y z +zx + 3xy, $$
$$ 5x^2 + 5y^2 + 8 z^2 +4zx +3xy, \; \; \; 5x^2 + 7 y^2 + 7 z^2 + 6 y z +zx + 5xy. $$ Cannot prove these relationships, of course. In my accounting, these are discriminant 232 and 648, respectively. The 232 genus has three classes, one more than I displayed, and the 648 genus has 6 classes. I'm just saying. For 232, the genus, collectively, misses $4n+2$ and $4^k (16n+6).$ These are traditionally called the "progressions." The two forms also miss $4^k \{1\}.$ For 648, the progressions are $9n \pm 3, \; \; 81n \pm 27, \;\; 4n+2, \; \; 4^k (16 n + 14).$ The two forms indicated above also miss $4^k \cdot \{1,40\}.$
NOTE, September 2017: I also checked the two pairs of forms I found for primitive representations. In both cases, the pair of forms in each genus primitively represent the same numbers up to a large bound. If ever proved, these examples violate Conjecture A on page 237, Regular Positive Ternary Quadratic Forms, J. S. Hsia, Mathematika, volume 28, (1981), pages 231-238.
Anyway, i have written a very fast program and am searching for other pairs. i am disappointed at finding so few.
Note that there are finiteness results about the number of exceptions missed by a ternary form, see Chan and Oh (2004), item 12 at OH_HOME. And, if we count number of representations, no two distinct forms match, i.e. the theta series determines the positive ternary form, proof A. Schiemann.
Meanwhile, note that, if we allow the discriminant to change, there are infinitely many pairs that represent the same numbers, elementary proofs:
$$ A(x^2 + xy+ y^2 ) + B z^2, \; \; \; A(x^2 + 3 y^2 ) + B z^2, $$
$$ A(x^2 + y^2 + z^2 ) + B (yz + z x + x y), \; \; \; Ax^2 + (2A-B) y^2 + (2A+B) z^2 + 2 B zx. $$ For this one, noticed by Irving Kaplansky, we have discriminant $\Delta = 4A^3 - 3 A B^2 + b^3 = (2A-B)^2 (A+B);$ for this to be positive definite, we need $A > 0$ and $2A > B > -A.$ Given that, the form still might not be "reduced," so it took a while to realize that the different appearances of this were really all the same.
Well, there is the QUESTION, are there only finitely many pairs of forms in the same genus that represent the same numbers?
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=====Discriminant 232 ==Genus Size== 3
Discriminant 232
Spinor genus misses no exceptions
232: 1 4 15 2 1 0 vs. s.g. 3 7 11 31 43
232: 3 3 7 1 2 1 vs. s.g. 1 single squareclass
232: 3 5 5 3 1 3 vs. s.g. 1 single squareclass
--------------------------size 3
The 150 smallest numbers represented by full genus
1 3 4 5 7 8 9 11 12 13
15 16 17 19 20 21 23 25 27 28
29 31 32 33 35 36 37 39 40 41
43 44 45 47 48 49 51 52 53 55
56 57 59 60 61 63 64 65 67 68
69 71 72 73 75 76 77 79 80 81
83 84 85 87 89 91 92 93 95 97
99 100 101 103 104 105 107 108 109 111
112 113 115 116 117 119 120 121 123 124
125 127 128 129 131 132 133 135 136 137
139 140 141 143 144 145 147 148 149 151
153 155 156 157 159 160 161 163 164 165
167 168 169 171 172 173 175 176 177 179
180 181 183 184 185 187 188 189 191 192
193 195 196 197 199 200 201 203 204 205
The 150 smallest numbers NOT represented by full genus
2 6 10 14 18 22 24 26 30 34
38 42 46 50 54 58 62 66 70 74
78 82 86 88 90 94 96 98 102 106
110 114 118 122 126 130 134 138 142 146
150 152 154 158 162 166 170 174 178 182
186 190 194 198 202 206 210 214 216 218
222 226 230 234 238 242 246 250 254 258
262 266 270 274 278 280 282 286 290 294
298 302 306 310 314 318 322 326 330 334
338 342 344 346 350 352 354 358 362 366
370 374 378 382 384 386 390 394 398 402
406 408 410 414 418 422 426 430 434 438
442 446 450 454 458 462 466 470 472 474
478 482 486 490 494 498 502 506 510 514
518 522 526 530 534 536 538 542 546 550
ALL ODD
Disc: 232
==================================
232: 1 4 15 2 1 0
misses, compared with full genus
3 7 11 12 28
31 43 44 48 56
79 112 115 120 124
141 165 168 172 176
184 192 224 295 301
309 316 448 456 460
471 480 487 496 555
564 568 589 616 660
672 688 704 736 760
768 805 840 896
232: 3 3 7 1 2 1
misses, compared with full genus
1: 1 2: 4 4: 16 8: 64 16: 256
ONE ODD
232: 3 5 5 3 1 3
misses, compared with full genus
1: 1 2: 4 4: 16 8: 64 16: 256
ONE ODD
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=====Discriminant 648 ==Genus Size== 6
Discriminant 648
Spinor genus misses no exceptions
648: 1 4 41 2 1 0 vs. s.g. 7 11 19 23 31
648: 1 5 35 4 1 1 vs. s.g. 8 13 29 31
648: 1 11 17 9 1 1 vs. s.g. 5 7 8 65 179
648: 4 5 9 3 0 2 vs. s.g. 1 8
648: 5 5 8 0 4 3 vs. s.g. 1 40
648: 5 7 7 6 1 5 vs. s.g. 1 40
--------------------------size 6
The 150 smallest numbers represented by full genus
1 4 5 7 8 9 11 13 16 17
19 20 23 25 28 29 31 32 35 36
37 40 41 43 44 45 47 49 52 53
55 59 61 63 64 65 67 68 71 72
73 76 77 79 80 81 83 85 88 89
91 92 95 97 99 100 101 103 104 107
109 112 113 115 116 117 119 121 124 125
127 128 131 133 136 137 139 140 143 144
145 148 149 151 152 153 155 157 160 161
163 164 167 169 171 172 173 175 176 179
180 181 185 187 188 191 193 196 197 199
200 203 205 207 208 209 211 212 215 217
220 221 223 225 227 229 232 233 235 236
239 241 243 244 245 247 251 252 253 256
257 259 260 261 263 265 268 269 271 272
The 150 smallest numbers NOT represented by full genus
2 3 6 10 12 14 15 18 21 22
24 26 27 30 33 34 38 39 42 46
48 50 51 54 56 57 58 60 62 66
69 70 74 75 78 82 84 86 87 90
93 94 96 98 102 105 106 108 110 111
114 118 120 122 123 126 129 130 132 134
135 138 141 142 146 147 150 154 156 158
159 162 165 166 168 170 174 177 178 182
183 184 186 189 190 192 194 195 198 201
202 204 206 210 213 214 216 218 219 222
224 226 228 230 231 234 237 238 240 242
246 248 249 250 254 255 258 262 264 266
267 270 273 274 276 278 282 285 286 290
291 294 297 298 300 302 303 306 309 310
312 314 318 321 322 326 327 330 334 336
Disc: 648
==================================
648: 1 4 41 2 1 0
misses, compared with full genus
7 11 19 23 28
31 35 44 76 79
88 92 107 112 124
140 152 176 280 304
316 344 352 368 428
448 472 496 536 560
608 616 704 811
648: 1 5 35 4 1 1
misses, compared with full genus
8 13 29 31 32
128 512
648: 1 11 17 9 1 1
misses, compared with full genus
5 7 8 32 65
128 179 512
648: 4 5 9 3 0 2
misses, compared with full genus
1: 1 8 32 128 512
648: 5 5 8 0 4 3
misses, compared with full genus
1: 1 2: 4 4: 16 40 8: 64
160 16: 256 640
648: 5 7 7 6 1 5
misses, compared with full genus
1: 1 2: 4 4: 16 40 8: 64
160 16: 256 640
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