11
$\begingroup$

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.

enter image description here

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?

=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= =====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 =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= =====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 =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
Improve this question
$\endgroup$
2
  • $\begingroup$ Bit of a relief to get a false positive overnight. Both discriminant $\Delta = 6804,$ not actually the same genus, one classically integral and the other not, and agreement up to my bound, $$ 32 \lfloor \sqrt[3] \Delta \rfloor, $$ partly because one form really does dominate the other, an observation by Kap in "Three Theorems" unpublished. This confirms that the software does recognize when two forms agree up to my bound. I was getting worried. $\endgroup$ Commented Dec 28, 2013 at 22:53
  • $\begingroup$ AHA! for some infinite families of pairs of forms, it is necessary to check up to, at least, something along the lines of $\Delta/12.$ So, i am just keeping the first test as it was, and then testing up to $100 + \Delta$ before having the computer record the pair. Also started a file of "false positives," pairs that agree up to the lower bound but not the higher. Onward and upward. $\endgroup$ Commented Dec 29, 2013 at 1:37

1 Answer 1

Reset to default
2
$\begingroup$

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. 

--------------------------------------------------------------- 54 : 2 2 4 1 2 0 216 : 2 4 8 4 1 1 75 : 1 4 5 1 1 0 111 : 1 4 7 1 0 0 78 : 3 3 3 1 1 3 142 : 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 3 190 : 3 5 5 5 2 3 142 : 3 3 5 2 3 1 158 : 3 3 5 -1 2 1 142 : 3 3 5 2 3 1 190 : 3 5 5 5 2 3 156 : 3 3 5 2 2 0 284 : 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 0 380 : 3 5 7 2 0 2 158 : 3 3 5 -1 2 1 190 : 3 5 5 5 2 3 162 : 2 2 14 1 2 2 648 : 2 6 14 3 1 0 177 : 2 4 7 4 2 1 213 : 2 4 7 0 1 1 225 : 3 4 7 4 3 3 333 : 3 4 7 1 0 0 232 : 3 3 7 1 2 1 232 : 3 5 5 3 1 3 284 : 3 5 6 4 2 2 316 : 3 5 6 0 2 2 284 : 3 5 6 4 2 2 380 : 3 5 7 2 0 2 316 : 3 5 6 0 2 2 380 : 3 5 7 2 0 2 324 : 4 4 6 0 3 2 567 : 4 6 7 3 2 3 486 : 2 2 41 1 2 2 1944 : 2 6 41 3 1 0 531 : 5 5 6 0 3 2 639 : 5 5 8 -1 2 4 648 : 4 7 7 5 2 2 2592 : 4 7 25 -4 2 2 648 : 5 5 8 0 4 3 648 : 5 7 7 6 1 5 675 : 5 5 8 -1 4 2 999 : 5 8 8 -5 1 4 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 .

=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=

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.

=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=

 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

=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=

Improve this answer
$\endgroup$
2
  • $\begingroup$ I don't understand your class. What is the meaning of to touch a computer all the decisions? If you find the solution then write the equation. Two forms are equated to each other. For a given condition and solve the equation. Turning the computer figures we do not understand the phenomenon. $\endgroup$ Commented Sep 25, 2017 at 6:46
  • $\begingroup$ @individ It may help you to read zakuski.math.utsa.edu/~kap/Timofeev_1963_Uspekhi.pdf The best explanation of this new material is this entire document zakuski.math.utsa.edu/~kap/Jagy_Encyclopedia.pdf especially pages 36-42. $\endgroup$ Commented Sep 25, 2017 at 17:42

You must log in to answer this question.

Start asking to get answers

Find the answer to your question by asking.

Ask question

Explore related questions

See similar questions with these tags.