https://michaelnielsen.org/polymath/index.php?action=history&feed=atom&title=Optimal_a-set_pairs 2026-05-08T02:14:54Z Revision history for this page on the wiki MediaWiki 1.42.3 https://michaelnielsen.org/polymath/index.php?title=Optimal_a-set_pairs&diff=1634&oldid=prev 2009-06-11T20:18:09Z

<p>New page: The row &quot;a1 a3 x * y = xy&quot; denotes the class of Level1, Level3 pairs, where Level1 is a 3D Moser set whose &quot;a&quot; points have signature a1 (when viewed as an 8-bit integer), and Level3 is a...</p> <p><b>New page</b></p><div>The row &quot;a1 a3 x * y = xy&quot; denotes the class of Level1, Level3 pairs, where Level1 is a 3D Moser set whose &quot;a&quot; points have signature a1 (when viewed as an 8-bit integer), and Level3 is a 3D Moser set whose &quot;a&quot; points have signature a3. x and y denote the number of 3D Moser sets with signature a1 and a3 respectively. There is one configuration per equivalence class (under the order 384 symmetry group of &lt;math&gt;[3]^4&lt;/math&gt;), and those configurations with a=1,2,3 have been deleted. Code for generating this list can be found [[Optimal a-set pair code|here]].<br /> <br /> 0 0 82173*82173 = 2457434633<br /> 0 15 82173*5750 = 472494750<br /> 0 23 82173*3071 = 252353283<br /> 0 27 82173*4021 = 330417633<br /> 0 30 82173*5000 = 410865000<br /> 0 31 82173*625 = 51358125<br /> 0 60 82173*9604 = 789189492<br /> 0 61 82173*784 = 64423632<br /> 0 63 82173*98 = 8052954<br /> 0 105 82173*4825 = 396484725<br /> 0 107 82173*512 = 42072576<br /> 0 111 82173*64 = 5259072<br /> 0 126 82173*64 = 5259072<br /> 0 127 82173*8 = 657384<br /> 0 255 82173*1 = 82173<br /> 1 22 82173*20472 = 1682245656<br /> 1 23 82173*3071 = 252353283<br /> 2 21 82173*16836 = 1383464628<br /> 2 22 82173*20472 = 1682245656<br /> 2 23 82173*3071 = 252353283<br /> 1 30 82173*5000 = 410865000<br /> 1 31 82173*625 = 51358125<br /> 1 44 82173*25777 = 2118173421<br /> 2 29 82173*4021 = 330417633<br /> 1 47 82173*625 = 51358125<br /> 1 60 82173*9604 = 789189492<br /> 1 61 82173*784 = 64423632<br /> 1 62 82173*784 = 64423632<br /> 1 63 82173*98 = 8052954<br /> 1 104 82173*20472 = 1682245656<br /> 1 105 82173*4825 = 396484725<br /> 1 106 82173*5000 = 410865000<br /> 1 107 82173*512 = 42072576<br /> 1 110 82173*784 = 64423632<br /> 1 111 82173*64 = 5259072<br /> 1 126 82173*64 = 5259072<br /> 1 127 82173*8 = 657384<br /> 8 19 82173*16836 = 1383464628<br /> 1 134 82173*20472 = 1682245656<br /> 8 23 82173*3071 = 252353283<br /> 1 137 82173*25777 = 2118173421<br /> 1 139 82173*4021 = 330417633<br /> 1 143 82173*625 = 51358125<br /> 1 150 82173*4825 = 396484725<br /> 1 151 82173*512 = 42072576<br /> 1 152 82173*25777 = 2118173421<br /> 1 153 82173*9604 = 789189492<br /> 1 154 82173*5000 = 410865000<br /> 1 155 82173*784 = 64423632<br /> 1 158 82173*512 = 42072576<br /> 1 159 82173*64 = 5259072<br /> 1 168 82173*16836 = 1383464628<br /> 1 169 82173*5000 = 410865000<br /> 1 170 82173*5750 = 472494750<br /> 1 171 82173*625 = 51358125<br /> 1 172 82173*4021 = 330417633<br /> 1 173 82173*784 = 64423632<br /> 1 174 82173*625 = 51358125<br /> 1 175 82173*98 = 8052954<br /> 1 188 82173*784 = 64423632<br /> 1 189 82173*64 = 5259072<br /> 1 190 82173*64 = 5259072<br /> 1 191 82173*8 = 657384<br /> 1 232 82173*3071 = 252353283<br /> 1 233 82173*512 = 42072576<br /> 1 234 82173*625 = 51358125<br /> 1 235 82173*64 = 5259072<br /> 1 238 82173*98 = 8052954<br /> 1 239 82173*8 = 657384<br /> 1 254 82173*8 = 657384<br /> 1 255 82173*1 = 82173<br /> 22 22 20472*20472 = 419102784<br /> 3 61 51060*784 = 40031040<br /> 3 63 51060*98 = 5003880<br /> 3 86 51060*5000 = 255300000<br /> 3 87 51060*625 = 31912500<br /> 6 52 51052*25777 = 1315967404<br /> 6 53 51052*4021 = 205280092<br /> 5 58 51060*4021 = 205312260<br /> 6 55 51052*625 = 31907500<br /> 3 94 51060*784 = 40031040<br /> 3 95 51060*98 = 5003880<br /> 3 104 51060*20472 = 1045300320<br /> 3 105 51060*4825 = 246364500<br /> 6 58 51052*4021 = 205280092<br /> 3 107 51060*512 = 26142720<br /> 3 108 51060*5000 = 255300000<br /> 3 109 51060*512 = 26142720<br /> 6 62 51052*784 = 40024768<br /> 3 111 51060*64 = 3267840<br /> 3 124 51060*784 = 40031040<br /> 3 125 51060*64 = 3267840<br /> 3 126 51060*64 = 3267840<br /> 3 127 51060*8 = 408480<br /> 3 192 51060*51060 = 2607123600<br /> 12 49 51060*16836 = 859646160<br /> 12 51 51060*5750 = 293595000<br /> 3 197 51060*4021 = 205312260<br /> 3 198 51060*5000 = 255300000<br /> 12 55 51060*625 = 31912500<br /> 3 207 51060*98 = 5003880<br /> 3 212 51060*3071 = 156805260<br /> 3 213 51060*625 = 31912500<br /> 3 214 51060*512 = 26142720<br /> 3 215 51060*64 = 3267840<br /> 3 216 51060*4021 = 205312260<br /> 3 217 51060*784 = 40031040<br /> 3 219 51060*64 = 3267840<br /> 3 220 51060*625 = 31912500<br /> 3 221 51060*98 = 5003880<br /> 3 222 51060*64 = 3267840<br /> 3 223 51060*8 = 408480<br /> 3 252 51060*98 = 5003880<br /> 3 253 51060*8 = 408480<br /> 3 255 51060*1 = 51060<br /> 6 96 51052*51052 = 2606306704<br /> 6 97 51052*20472 = 1045136544<br /> 20 42 51052*16836 = 859511472<br /> 20 43 51052*3071 = 156780692<br /> 21 42 16836*16836 = 283450896<br /> 6 103 51052*784 = 40024768<br /> 6 105 51052*4825 = 246325900<br /> 6 107 51052*512 = 26138624<br /> 6 111 51052*64 = 3267328<br /> 6 114 51052*4021 = 205280092<br /> 6 115 51052*625 = 31907500<br /> 6 118 51052*784 = 40024768<br /> 6 119 51052*98 = 5003096<br /> 6 120 51052*5000 = 255260000<br /> 6 121 51052*512 = 26138624<br /> 6 122 51052*784 = 40024768<br /> 6 123 51052*64 = 3267328<br /> 6 126 51052*64 = 3267328<br /> 6 127 51052*8 = 408416<br /> 6 144 51052*51052 = 2606306704<br /> 9 97 51052*20472 = 1045136544<br /> 6 147 51052*5000 = 255260000<br /> 6 150 51052*4825 = 246325900<br /> 6 151 51052*512 = 26138624<br /> 6 159 51052*64 = 3267328<br /> 6 176 51052*16836 = 859511472<br /> 6 177 51052*4021 = 205280092<br /> 6 178 51052*3071 = 156780692<br /> 6 179 51052*625 = 31907500<br /> 6 180 51052*5000 = 255260000<br /> 6 181 51052*784 = 40024768<br /> 6 182 51052*512 = 26138624<br /> 6 183 51052*64 = 3267328<br /> 6 185 51052*784 = 40024768<br /> 6 187 51052*98 = 5003096<br /> 6 189 51052*64 = 3267328<br /> 6 191 51052*8 = 408416<br /> 6 240 51052*5750 = 293549000<br /> 6 241 51052*625 = 31907500<br /> 6 242 51052*625 = 31907500<br /> 6 243 51052*98 = 5003096<br /> 6 246 51052*64 = 3267328<br /> 6 247 51052*8 = 408416<br /> 6 249 51052*64 = 3267328<br /> 6 251 51052*8 = 408416<br /> 6 255 51052*1 = 51052<br /> 7 118 16836*784 = 13199424<br /> 7 119 16836*98 = 1649928<br /> 7 120 16836*5000 = 84180000<br /> 7 121 16836*512 = 8620032<br /> 7 122 16836*784 = 13199424<br /> 7 123 16836*64 = 1077504<br /> 7 126 16836*64 = 1077504<br /> 7 127 16836*8 = 134688<br /> 7 176 16836*16836 = 283450896<br /> 11 113 16836*3071 = 51703356<br /> 24 63 69868*98 = 6847064<br /> 11 116 16836*4021 = 67697556<br /> 11 117 16836*625 = 10522500<br /> 7 182 16836*512 = 8620032<br /> 7 183 16836*64 = 1077504<br /> 7 188 16836*784 = 13199424<br /> 7 189 16836*64 = 1077504<br /> 7 191 16836*8 = 134688<br /> 7 224 16836*16836 = 283450896<br /> 14 113 16836*3071 = 51703356<br /> 7 226 16836*4021 = 67697556<br /> 14 115 16836*625 = 10522500<br /> 7 230 16836*784 = 13199424<br /> 7 231 16836*64 = 1077504<br /> 7 233 16836*512 = 8620032<br /> 7 235 16836*64 = 1077504<br /> 7 239 16836*8 = 134688<br /> 7 240 16836*5750 = 96807000<br /> 7 241 16836*625 = 10522500<br /> 7 242 16836*625 = 10522500<br /> 7 243 16836*98 = 1649928<br /> 7 246 16836*64 = 1077504<br /> 7 247 16836*8 = 134688<br /> 7 248 16836*625 = 10522500<br /> 7 249 16836*64 = 1077504<br /> 7 250 16836*98 = 1649928<br /> 7 251 16836*8 = 134688<br /> 7 254 16836*8 = 134688<br /> 7 255 16836*1 = 16836<br /> 15 240 5750*5750 = 33062500<br /> 15 241 5750*625 = 3593750<br /> 15 243 5750*98 = 563500<br /> 15 246 5750*64 = 368000<br /> 15 247 5750*8 = 46000<br /> 15 255 5750*1 = 5750<br /> 22 104 20472*20472 = 419102784<br /> 22 105 20472*4825 = 98777400<br /> 23 104 3071*20472 = 62869512<br /> 22 107 20472*512 = 10481664<br /> 23 106 3071*5000 = 15355000<br /> 22 111 20472*64 = 1310208<br /> 22 126 20472*64 = 1310208<br /> 22 127 20472*8 = 163776<br /> 22 129 20472*69868 = 1430337696<br /> 24 99 69868*5000 = 349340000<br /> 22 134 20472*20472 = 419102784<br /> 24 103 69868*784 = 54776512<br /> 24 105 69868*4825 = 337113100<br /> 24 107 69868*512 = 35772416<br /> 24 110 69868*784 = 54776512<br /> 24 111 69868*64 = 4471552<br /> 22 150 20472*4825 = 98777400<br /> 22 151 20472*512 = 10481664<br /> 22 152 20472*25777 = 527706744<br /> 25 105 25777*4825 = 124374025<br /> 22 154 20472*5000 = 102360000<br /> 25 107 25777*512 = 13197824<br /> 22 158 20472*512 = 10481664<br /> 22 159 20472*64 = 1310208<br /> 22 169 20472*5000 = 102360000<br /> 22 171 20472*625 = 12795000<br /> 22 172 20472*4021 = 82317912<br /> 26 109 25777*512 = 13197824<br /> 22 174 20472*625 = 12795000<br /> 26 111 25777*64 = 1649728<br /> 22 188 20472*784 = 16050048<br /> 22 189 20472*64 = 1310208<br /> 22 190 20472*64 = 1310208<br /> 22 191 20472*8 = 163776<br /> 22 233 20472*512 = 10481664<br /> 22 234 20472*625 = 12795000<br /> 22 235 20472*64 = 1310208<br /> 22 238 20472*98 = 2006256<br /> 22 239 20472*8 = 163776<br /> 22 254 20472*8 = 163776<br /> 22 255 20472*1 = 20472<br /> 23 126 3071*64 = 196544<br /> 23 127 3071*8 = 24568<br /> 23 129 3071*69868 = 214564628<br /> 24 115 69868*625 = 43667500<br /> 24 119 69868*98 = 6847064<br /> 24 121 69868*512 = 35772416<br /> 24 123 69868*64 = 4471552<br /> 24 126 69868*64 = 4471552<br /> 24 127 69868*8 = 558944<br /> 23 150 3071*4825 = 14817575<br /> 23 151 3071*512 = 1572352<br /> 23 152 3071*25777 = 79161167<br /> 25 121 25777*512 = 13197824<br /> 23 154 3071*5000 = 15355000<br /> 25 123 25777*64 = 1649728<br /> 23 158 3071*512 = 1572352<br /> 23 159 3071*64 = 196544<br /> 26 121 25777*512 = 13197824<br /> 26 123 25777*64 = 1649728<br /> 23 172 3071*4021 = 12348491<br /> 26 125 25777*64 = 1649728<br /> 26 126 25777*64 = 1649728<br /> 26 127 25777*8 = 206216<br /> 23 188 3071*784 = 2407664<br /> 23 189 3071*64 = 196544<br /> 23 190 3071*64 = 196544<br /> 23 191 3071*8 = 24568<br /> 23 232 3071*3071 = 9431041<br /> 23 233 3071*512 = 1572352<br /> 23 234 3071*625 = 1919375<br /> 23 235 3071*64 = 196544<br /> 23 238 3071*98 = 300958<br /> 23 239 3071*8 = 24568<br /> 23 254 3071*8 = 24568<br /> 23 255 3071*1 = 3071<br /> 24 231 69868*64 = 4471552<br /> 24 239 69868*8 = 558944<br /> 24 255 69868*1 = 69868<br /> 25 225 25777*5000 = 128885000<br /> 25 227 25777*784 = 20209168<br /> 25 230 25777*784 = 20209168<br /> 25 231 25777*64 = 1649728<br /> 25 233 25777*512 = 13197824<br /> 25 234 25777*625 = 16110625<br /> 25 235 25777*64 = 1649728<br /> 25 238 25777*98 = 2526146<br /> 25 239 25777*8 = 206216<br /> 25 241 25777*625 = 16110625<br /> 25 243 25777*98 = 2526146<br /> 25 246 25777*64 = 1649728<br /> 25 247 25777*8 = 206216<br /> 25 248 25777*625 = 16110625<br /> 25 249 25777*64 = 1649728<br /> 47 92 625*4021 = 2513125<br /> 25 251 25777*8 = 206216<br /> 25 254 25777*8 = 206216<br /> 25 255 25777*1 = 25777<br /> 27 214 4021*512 = 2058752<br /> 27 215 4021*64 = 257344<br /> 27 216 4021*4021 = 16168441<br /> 46 117 4021*625 = 2513125<br /> 27 219 4021*64 = 257344<br /> 27 222 4021*64 = 257344<br /> 27 223 4021*8 = 32168<br /> 27 228 4021*4021 = 16168441<br /> 45 121 5000*512 = 2560000<br /> 27 231 4021*64 = 257344<br /> 27 236 4021*625 = 2513125<br /> 27 237 4021*64 = 257344<br /> 30 190 5000*64 = 320000<br /> 27 239 4021*8 = 32168<br /> 27 252 4021*98 = 394058<br /> 27 253 4021*8 = 32168<br /> 27 255 4021*1 = 4021<br /> 30 225 5000*5000 = 25000000<br /> 61 105 784*4825 = 3782800<br /> 30 230 5000*784 = 3920000<br /> 30 231 5000*64 = 320000<br /> 30 233 5000*512 = 2560000<br /> 30 235 5000*64 = 320000<br /> 30 238 5000*98 = 490000<br /> 30 239 5000*8 = 40000<br /> 30 241 5000*625 = 3125000<br /> 61 109 784*512 = 401408<br /> 30 246 5000*64 = 320000<br /> 30 247 5000*8 = 40000<br /> 30 249 5000*64 = 320000<br /> 31 234 625*625 = 390625<br /> 30 251 5000*8 = 40000<br /> 30 254 5000*8 = 40000<br /> 30 255 5000*1 = 5000<br /> 31 241 625*625 = 390625<br /> 31 242 625*625 = 390625<br /> 61 125 784*64 = 50176<br /> 31 246 625*64 = 40000<br /> 31 247 625*8 = 5000<br /> 31 248 625*625 = 390625<br /> 31 249 625*64 = 40000<br /> 62 126 784*64 = 50176<br /> 31 251 625*8 = 5000<br /> 31 254 625*8 = 5000<br /> 31 255 625*1 = 625<br /> 105 105 4825*4825 = 23280625<br /> 105 107 4825*512 = 2470400<br /> 107 107 512*512 = 262144<br /> 105 111 4825*64 = 308800<br /> 107 109 512*512 = 262144<br /> 107 111 512*64 = 32768<br /> 111 111 64*64 = 4096<br /> 105 126 4825*64 = 308800<br /> 105 127 4825*8 = 38600<br /> 62 214 784*512 = 401408<br /> 107 125 512*64 = 32768<br /> 107 126 512*64 = 32768<br /> 107 127 512*8 = 4096<br /> 61 237 784*64 = 50176<br /> 111 123 64*64 = 4096<br /> 61 253 784*8 = 6272<br /> 111 126 64*64 = 4096<br /> 111 127 64*8 = 512<br /> 126 126 64*64 = 4096<br /> 126 127 64*8 = 512<br /> 127 127 8*8 = 64<br /> 105 150 4825*4825 = 23280625<br /> 105 151 4825*512 = 2470400<br /> 107 151 512*512 = 262144<br /> 107 159 512*64 = 32768<br /> 111 159 64*64 = 4096<br /> 107 189 512*64 = 32768<br /> 107 191 512*8 = 4096<br /> 107 214 512*512 = 262144<br /> 107 215 512*64 = 32768<br /> 107 223 512*8 = 4096<br /> 107 253 512*8 = 4096<br /> 107 255 512*1 = 512<br /> 111 246 64*64 = 4096<br /> 111 247 64*8 = 512<br /> 111 249 64*64 = 4096<br /> 111 251 64*8 = 512<br /> 111 255 64*1 = 64<br /> 126 255 64*1 = 64<br /> 127 254 8*8 = 64<br /> 127 255 8*1 = 8<br /> 255 255 1*1 = 1</div>

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