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2019-12-27, 09:15
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#1 |
Jun 2015
Vallejo, CA/.
2×3×191 Posts |
QUADRUPLET PRIMES AND THEIR GAPS
Here is a list of first instances For the Quadruplets (notice how different it is from the list of First Differences between Twin Primes)
But first a review GAPS between primes P≥3 Minumum gasp is 2 Maximum gap Until Dec 2019 is the gap of 5,103,138 after a PRP of 216,849 digits. GAP between Twins p,p+2 to q,q+2 these gaps between p and q are always of the form 6k Minimum gap (expressed as k) 1 Maximum gap yet found 518546 As to the gap between Quadruplet primes we define the number k as the distance between the first element of the QP (or alternatively of the barycenter) divided by 30 The barycenter –or Pivot– is the central number (it ends in 5) For instance the baricenter of the quadruplet {821, 823, 827, 829 is 825 A159910 FIRST OCCURRENCE Code:
Gap After p= 1 1006301 rudy235 2 NONE 3 11 rudy235 4 1022381 rudy235 5 NONE 6 3512051 rudy235 7 1871 rudy235 8 632081 rudy235 9 NONE 10 71121831 rudy235 11 15731 rudy235 12 NONE 13 1481 rudy235 14 1155611 rudy235 15 1068251 rudy235 16 NONE 17 18911 rudy235 18 284741 rudy235 19 NONE 20 12390011 rudy235 21 191 rudy235 22 821 rudy235 23 NONE 24 3837131 rudy235 25 875261 rudy235 26 NONE 27 854921 rudy235 28 10865291 rudy235 29 18041 rudy235 30 NONE 31 958541 rudy235 32 680291 rudy235 33 NONE 34 299471 rudy235 35 1063961 rudy235 36 3995441 rudy235 37 NONE 38 4404551 39 5733525 40 NONE 3'798,075-3,741,165= 56910 . 56,910/30 = 1897 |
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2019-12-27, 12:06
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#2 |
Dec 2008
you know...around...
24×53 Posts |
You might want to have a look at this paper, which lists record gaps between prime quadruplets up to 10^15:
https://arxiv.org/abs/1309.4053 |
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2019-12-27, 14:53
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#3 | |
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Jun 2015
Vallejo, CA/.
2×3×191 Posts |
Quote:
Here is a more complete table of First Occurrences. (It is easy to prove GAPS 2 mod 7 and 5 mod 7 do not exist. There were a couple of mistakes in the previous table This is the corrected version Note gaps like 2,5,9,12,16,19,23,26,30,33,37,40,44,47,51,54,58,61,65,68,72,75,79,82,86,89,93,96,100… and all 2 mod7 or 5 mod cannot exist. Code:
GAP After Quadruple 1 1006301 3 11 4 1246241 6 3512051 7 1871 8 632081 10 1121831 11 15731 13 1481 14 1155611 15 1068251 17 18911 18 284741 20 12390011 21 191 22 821 24 3837131 25 875261 27 854921 28 10865291 29 18041 31 958541 32 680291 34 299471 35 1063961 36 663581 38 165701 39 2081 41 Unknown 42 21011 43 97841 45 79691 46 2223671 48 1022501 49 3436241 50 300491 52 294311 53 19421 55 Unknown 56 81041 57 463451 59 442571 60 419051 62 1902611 63 734471 64 6054281 66 16061 67 6332861 69 10037381 70 2246141 71 266681 73 3461 74 855731 76 67211 77 358901 78 7159511 80 2761721 81 77261 83 661091 84 1006331 85 257861 87 467471 88 13001 90 389561 91 69491 92 116531 94 2846861 95 594821 97 119291 98 1003361 99 11718821 101 22271 102 1230371 104 31721 105 2728541 106 1830341 108 1065011 109 5154761 111 654161 112 833711 113 Unknown 115 1627601 116 1912451 118 9585881 119 9431 120 295871 122 240041 123 5248091 125 22692611 126 5651 127 1023941 129 6880871 130 243701 132 2875331 133 51341 134 1798631 136 8557721 137 1588751 139 5201291 140 559211 141 62981 143 2541941 144 166841 146 397751 147 6556601 148 8827541 150 5234681 151 1122131 153 4575281 154 495611 155 354251 157 741751 158 2418671 160 Unknown 161 768191 162 13179581 164 1616801 165 1954361 167 Unknown 168 72221 169 19842671 171 1324571 172 Unknown 174 Unkown 175 2413421 176 2002331 178 14912741 179 978071 181 Unknown 182 340931 183 Unknown 185 14877671 186 Unknown 188 1791731 189 2136131 190 1508621 192 195731 193 1904471 195 6187451 196 1157831 197 Unknown 199 4734671 200 823721 The largest gap I've found is 5261 after 21872441 |
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2019-12-27, 16:46
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#4 |
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Jun 2015
Vallejo, CA/.
2×3×191 Posts |
More First occurrences
Code:
GAP After Quadruple 202 12021671 203 82721 204 13597271 206 16728071 207 4326401 209 260411 210 4609301 211 1749491 213 420851 214 25301 216 Unknown 217 334421 218 7267451 220 1352201 221 563411 223 109841 224 4138241 225 8101781 227 6918341 228 1093991 230 18468251 231 470081 232 15010571 234 412031 235 1711811 237 9541991 238 1577291 239 7328771 241 187631 242 4032401 244 Unknown 245 6692261 246 12382631 248 3727721 249 2704901 251 Unknown 252 43781 253 4370081 255 55331 256 1360781 258 10872131 259 4396781 260 9291641 262 9725951 263 4405691 265 Unknown 266 217361 267 13056311 269 2596661 270 814061 272 4353311 273 444341 274 14235671 276 326141 277 946661 279 5417381 280 881471 281 157271 283 2007911 284 11138711 286 11192801 287 938051 288 Unknown 290 135461 291 101111 294 1146791 295 9138131 297 585911 298 34841 300 5318891 301 88811 302 Unknown 304 3215741 305 19451981 307 5037911 308 6474821 309 402761 311 2604731 312 10606361 314 2159231 315 6631901 316 9617981 318 2294051 321 597671 322 539501 323 12600821 325 1514321 326 Unknown 328 13680671 329 247991 330 7465481 332 5841461 333 705161 335 549161 336 20612231 337 Unknown 339 Unknown 340 14974571 342 2065571 343 9154961 344 Unknown 346 500231 347 Unknown 349 681251 350 1819841 351 5425751 353 Unknown 354 803441 356 Unknown 357 1246361 358 Unknown 360 7659011 361 7422221 363 5158031 364 452531 365 4144961 367 Unknown 368 8183891 370 9280541 371 2362271 372 Unknown 374 1126661 375 1780481 377 4174601 378 Unknown 379 Unknown 381 21633371 382 1659101 384 8658101 385 6715991 386 Unknown 388 Unknown 389 2050331 391 20598701 392 12027731 393 Unknown 395 11477231 396 4098461 396 5450111 398 12953651 399 3328901 400 Unknown |
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2020-01-01, 17:27
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#5 |
May 2018
23×5×7 Posts |
The value of the gap of 4 should be 1022381.
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2020-01-01, 18:14
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#6 | |
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Jun 2015
Vallejo, CA/.
2×3×191 Posts |
Quote:
Can you find the first occurrence on 41? I have searched up to 5,001,000 to no avail. 
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2020-01-01, 20:54
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#7 |
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Jun 2015
Vallejo, CA/.
100011110102 Posts |
I have found the first occurrences of many of the missing gaps.
Code:
GAP after prime 41 181773281 55 72506501 113 51633731 160 78139751 167 46187831 172 45290801 174 23482001 181 33113651 |
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2020-01-02, 15:02
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#8 |
Feb 2003
78016 Posts |
Here is a C program based on my twingaps program (itself based on fastsieve by TOS). A Windows binary is included as well as a batch file for running it.
In order to save the results you should redirect the output into a text file, e.g.: Code:
run.bat >> logfile.txt BTW.: It runs pretty fast. Up to 1 billion (1e9) in just 1 second...  And a list of largest gaps found up to 1e9: Code:
10045 948653801 10074 785474531 10164 736611521 10409 540053531 10415 783970331 10515 511502861 10528 204335771 10780 898511981 11133 737220431 11141 987996431 11200 644979191 12417 643002791 13184 628246181 |
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2020-01-02, 16:35
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#9 |
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Feb 2003
27×3×5 Posts |
Here are the first occurrences of gaps between quadruplets searched up to 1e12.
Largest gap: 85025 342611795411 Smallest unknown gap: 21764 I stopped my search at 1e12, so you may continue from there... |
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2020-01-02, 21:25
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#10 | |
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Jun 2015
Vallejo, CA/.
2·3·191 Posts |
Quote:
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2020-01-03, 13:19
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#11 | |
Jun 2003
Suva, Fiji
23·3·5·17 Posts |
Quote:
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