.LOG Generalized Hyper-Cullen primes form k^n*n^k+1, k>n all ns greater than 11 and less than 100 checked to at least 25000 all ns less than 400 checked to at least 15000 all ns less than 500 checked to at least 10000 all ns less than 2048 checked to at least 3000 listed as "n k(s) [limit]" extra thanks to Mark Rodenkirch for his Multisieve program n k 2 3, 4, 21, 30, 33, 57, 100, 142, 144, 150, 198, 225, 304, 513, 782, 858, 3638, 6076, 9297, 11037, 12135(Lifchitz, g2, 1997), 12876(Lifchitz, g2, 1997), 30180(Lifchitz, g2, 1997), 48470(Oakes, p42, 2000) [200000](Harvey) 3 20, 110, 152 [100000] (Blazek) 4 6, 10, 12, 190, 382, 1218, 2300, 6072, 25844 [32750] 49518(Kazuyoshi, p14, 2001) 5 154, 1818, 12948 [30000] 6 16, 20, 46, 94, 2440, 7600 [16384] 8 28, 104, 116, 476, 2012, 3420 [16384] 9 20, 1640 [20000] 10 16, 28, 576, 759, 2114, 2256, 4774(Hugo Platzer 10-27-6) [10000] 11 1218 [25000] 12 20, 28, 196, 2860, 6428 14 15, 254, 926, 1276, 1950, 3004 15 15458 16 22, 46, 50, 153, 13774 17 802 18 316, 1664, 2413 20 33, 36, 192, 6867, 10696 21 38 22 23, 30, 38, 322, 12840 24 250, 1814, 2275, 2338, 8194, 15200 25 13504 26 36, 46, 230, 320, 666, 8130 28 30, 584, 928, 2372, 4870 29 1468, 20400 30 122, 2576 32 1568, 8145, 11920 33 1312 34 65, 550, 3080, 4816 35 4696, 17692 36 12622, 19952 37 18472 38 162 39 746 40 84 41 516 42 124 43 798 44 104, 328, 3724 45 3898 46 390, 524 48 91, 455, 8192 49 24580 50 56, 19704 51 110, 898 52 68 54 134, 143, 224, 230, 523, 9343 55 24474 56 81, 88, 687 58 1264, 3050 60 592 62 610, 12243 63 5156, 20756 64 740 67 12654 68 404, 440, 4851 70 92, 15786 71 220, 16716 72 104 75 1064, 15404 76 3970, 5659, 17230 78 218 80 1108, 7774, 9276 82 790, 7342, 15600, 16676 83 3294 84 194, 278 85 2772 86 7174 87 9214, 11752 88 568, 1264, 21150 90 16202 91 1198 92 1336, 5830 93 242 95 6538, 11730 96 3772 97 220 98 144, 822, 1884, 18996, 23596 100 614, 1924 102 302, 5420 104 114, 272, 420, 15874 106 164 110 597, 756 112 268, 2148 113 15702 114 944, 2723, 5386 115 16612 116 192, 12782, 18570 118 3818 120 121, 3034, 3842 122 5720 123 596 124 1324, 3208 126 430 128 4616 130 9084 131 706 132 4670, 8720 134 543 136 858 139 2188, 16198 140 3644 142 760, 907, 2825, 3881 144 1352 145 504 146 1394, 2966 147 416 148 1052, 2050, 10880 152 272 154 2540, 14995 156 6974 158 8840 160 1198, 4464 161 2928 162 802, 3022 164 422, 1515 165 266, 14126 169 2560 170 1016, 2853, 5554, 11502 171 622, 1468 172 2880, 5284 173 220 184 9218 186 491 187 1614, 3772 188 384, 3688 189 12616 190 7492 192 1456, 3281 197 7836 202 236, 6964 204 740, 2464, 3448, 3884 207 1820, 10294 208 885, 3335, 5136 209 582 212 2776 ,12196 213 7300 214 4340, 16955 219 2492 220 1414, 3679 225 11204 227 11194 228 5968 232 564, 5192 234 265, 7463, 9010 236 2979 238 19384 240 12866 242 1340 244 1108 246 4784 248 1305, 3356 250 2254, 7448 251 4350 252 2647 254 340, 754, 1473,9080 255 9398 256 566, 784, 6746, 8970 , 13318 257 15048 260 2854, 3868, 10968 264 475 265 6746, 5094 268 668 270 412 271 4350 273 7460 274 4616, 6074 277 2008 278 764, 2049 282 6430 284 630, 3264 286 1055 288 680, 1676, 7904 290 10593 291 7280 292 879 293 8358 294 1304 296 356, 1202, 1898, 3058, 7084 298 9922 299 10818 300 15323 310 446 315 1448, 2966, 4472 316 5990 317 9898 318 556 320 1132, 1232 322 453, 2215, 5982 324 686, 9610 325 954, 3712 326 516 328 719, 1940, 3923 330 6236, 12238 332 724, 13812 334 14067 335 684 339 14332 341 8392 343 1518 346 9470 348 452 354 1930 358 1644, 2330, 8604 363 8318 364 6236 365 472 367 2982 368 1209 376 386, 1133 377 5770 380 5226, 5524 385 2074, 4572 386 5180, 9009 388 1336 392 424 394 3910, 6707 395 9084 396 1270, 9418 399 716 400 1112 401 7980 404 1384 406 3856 407 4768 412 6828 414 3395 415 1638 416 2130 420 544, 5678 422 1041 428 852, 9740 430 1972, 3922 431 1632, 5998, 7390 434 9854 436 862, 4475 437 1638 440 1546 446 510 447 662 448 564 454 1180, 1446 464 476 465 7538 468 3610 470 1202 472 1580 , 5661 474 5434, 6616 476 8528 484 9554 486 1786 488 1210 489 7460 490 1688, 3256, 3408 492 844, 3944 493 2122 496 5541 498 6076 500 2396 502 2040, 2456 508 1152 509 1908 510 1936 512 2439 514 663 516 1150 518 1522 521 2050 522 1900 524 1908 536 554 538 789 540 2122 544 785 546 2594 554 2104 555 988, 2678 556 2554 572 747, 2240 580 822, 1606 582 1844 584 1598 585 776 586 2714 588 661 600 1748 604 2332 610 2026 620 952, 2626 626 2456 628 859 632 1488 644 1128 650 748, 1414 657 1292 670 886 674 1320 680 1896 686 1126 688 1370 692 2120 704 916 710 2306 715 2322 718 2370 720 962 724 1988 728 2364 732 1280 740 2301 744 1015 746 2006 748 1729 752 984 758 939 764 2928 776 2562 778 917, 2462 787 2914 796 1490 801 1022 802 899 806 2396 816 2872 832 968 836 1874 853 868 855 1654 858 2536 862 1576 864 2945 890 1254, 1902 908 1800 914 2220 936 1670 961 1902 966 2090 974 1814 982 1476 984 1196 994 1256 1004 2420 1018 1556 1031 2326 1036 2201 1044 2650 1046 2979 1059 2498 1072 1729, 2364 1082 1761 1082 1761 1132 1624 1174 2136 1206 2464 1230 2012 1258 2376 1260 1766 1332 1729 1334 2316 1336 2970 1342 1677 1348 2079 1358 1647 1376 2360 1434 2420 1446 1750 1476 2513 1496 2020 1500 1771 1534 1673 1550 2686 1570 1616 1575 1894 1606 1836 1632 2852 1642 2343 1652 2216 1658 2079 1676 2434 1678 2260 1684 1942 1710 1834 1711 2512 1736 1910 1740 2854, 2912 1810 1816 1820 2668 1838 2097 1850 2511 1866 1984 1877 2058 1893 1994 1934 2146 1940 2066 1950 2306 1954 2746 1972 2120 2002 2690 2016 2990 4000 4414 (Ian Brown, p90, 11-19-2) 30479 digits [20000] 4007 4074 (Ian Brown, p90, 11-20-2) 29144 digits [20000] 4022 6195 (Ian Brown, p90, 9-30-2) 37582 digits [20000] 4032 5230 (Ian Brown, p90, 9-16-2) 33850 digits [20000] 4057 8380 (Ian Brown, p90, 10-10-2) 46154 digits [20000] 4062 10379 (Ian Brown, p90, 10-10-2) 53769 digits [20000] News: 6/18/13 Herranen checks b=401-500, 3001-10000, finds 7980^401*401^7980+1 3856^406*406^3856+1 4768^407*407^4768+1 6828^412*412^6828+1 3395^414*414^3395+1 5678^420*420^5678+1 9740^428*428^9740+1 3922^430*430^3922+1 5998^431*431^5998+1 7390^431*431^7390+1 9854^434*434^9854+1 4475^436*436^4475+1 7538^465*465^7538+1 3610^468*468^3610+1 5661^472*472^5661+1 5434^474*474^5434+1 6616^474*474^6616+1 8528^476*476^8528+1 9554^484*484^9554+1 4760^489*489^4760+1 3256^490*490^3256+1 3408^490*490^3408+1 3944^492*492^3944+1 5541^496*496^5541+1 6076^498*498^6076+1 6/13/13 Herranen checks b= 201 to 300 , 15-20k, finds 16955^214*214^16955+1 19384^238*238^19384+1 15048^257*257^15048+1 15323^300*300^15323+1 6/11/13 Herranen checks b= 301 to 400, 10-15k, finds 12238^330*330^12238+1 13812^332*332^13812+1 14067^334*334^14067+1 14332^339*339^14332+1 6/3/13 Herranen checks b=301 to 400, 3001 to 10k, finds 4472^315*315^4472+1 5990^316*316^5990+1 9898^317*317^9898+1 5982^322*322^5982+1 9610^324*324^9610+1 3712^325*325^3712+1 3923^328*328^3923+1 6236^330*330^6236+1 8392^341*341^8392+1 9470^346*346^9470+1 8604^358*358^8604+1 8318^363*363^8318+1 6236^364*364^6236+1 5770^377*377^5770+1 5226^380*380^5226+1 5524^380*380^5524+1 4572^385*385^4572+1 5180^386*386^5180+1 9009^386*386^9009+1 3910^394*394^3910+1 6707^394*394^6707+1 9084^395*395^9084+1 9418^396*396^9418+1 5/27/13 Herranen checks b= 201-300 10-15k, finds 10294^207*207^10294+1 12196^212*212^12196+1 11204^225*225^11204+1 11194^227*227^11194+1 12866^240*240^12866+1 13318^256*256^13318+1 10968^260*260^10968+1 10593^290*290^10593+1 10818^299*299^10818+1 and b=101 to 200 , 15k-20k, finds 15874^104*104^15874+1 15702^113*113^15702+1 16612^115*115^16612+1 18570^116*116^18570+1 16198^139*139^16198+1 5/17/13 Herranen checks b=101 to 200, 10000 to 15000 Found: 12782^116*116^12782+1 10880^148*148^10880+1 14995^154*154^14995+1 14126^165*165^14126+1 11502^170*170^11502+1 12616^189*189^12616+1 also b=201 to 300, 3000 to 10000 Found: 6964^202*202^6964+1 3448^204*204^3448+1 3884^204*204^3884+1 3335^208*208^3335+1 5136^208*208^5136+1 7300^213*213^7300+1 4340^214*214^4340+1 3679^220*220^3679+1 5968^228*228^5968+1 5192^232*232^5192+1 7463^234*234^7463+1 9010^234*234^9010+1 4784^246*246^4784+1 3356^248*248^3356+1 7448^250*250^7448+1 4350^251*251^4350+1 9080^254*254^9080+1 9398^255*255^9398+1 6746^256*256^6746+1 8970^256*256^8970+1 3868^260*260^3868+1 5094^265*265^5094+1 4350^271*271^4350+1 7460^273*273^7460+1 4616^274*274^4616+1 6074^274*274^6074+1 6430^282*282^6430+1 3264^284*284^3264+1 7904^288*288^7904+1 7280^291*291^7280+1 8358^293*293^8358+1 3058^296*296^3058+1 7084^296*296^7084+1 9922^298*298^9922+1 12/29/13 Herranen checks b=11 to 100, up to 25000; and b=101 to 200, up to 10000. Found: 4870^28*28^4870+1 4816^34*34^4816+1 4696^35*35^4696+1 4851^68*68^4851+1 15458^15*15^15458+1 15200^24*24^15200+1 17692^35*35^17692+1 19952^36*36^19952+1 18472^37*37^18472+1 19704^50*50^19704+1 15786^70*70^15786+1 16716^71*71^16716+1 15404^75*75^15404+1 17230^76*76^17230+1 15600^82*82^15600+1 16676^82*82^16676+1 16202^90*90^16202+1 18996^98*98^18996+1 6428^12*12^6428+1 3004^14*14^3004+1 6867^20*20^6867+1 3080^34*34^3080+1 3724^44*44^3724+1 3898^45*45^3898+1 3050^58*58^3050+1 3970^76*76^3970+1 3294^83*83^3294+1 3772^96*96^3772+1 8194^24*24^8194+1 8130^26*26^8130+1 8145^32*32^8145+1 8192^48*48^8192+1 9343^54*54^9343 +1 5156^63*63^5156+1 5659^76*76^5659+1 7774^80*80^7774+1 9276^80*80^9276+1 7342^82*82^7342+1 7174^86*86^7174+1 9214^87*87^9214+1 5830^92*92^5830+1 6538^95*95^6538+1 20400^29*29^20400+1 24580^49*49^24580+1 24474^55*55^24474+1 20756^63*63^20756+1 21150^88*88^21150+1 23596^98*98^23596+1 13774^16*16^13774+1 10696^20*20^10696+1 12840^22*22^12840+1 13504^25*25^13504+1 11920^32*32^11920+1 12622^36*36^12622+1 12243^62*62^12243+1 12654^67*67^12654+1 11752^87*87^11752+1 11730^94*94^11730+1 5420^102*102^5420+1 5386^114*114^5386+1 3818^118*118^3818+1 3034^120*120^3034+1 3842^120*120^3842+1 5720^122*122^5720+1 3208^124*124^3208+1 4616^128*128^4616+1 9084^130*130^9084+1 4670^132*132^4670+1 8720^132*132^8720+1 3644^140*140^3644+1 3881^142*142^3881+1 6974^156*156^6974+1 8840^158*158^8840+1 4464^160*160^4464+1 3022^162*162^3022+1 5554^170*170^5554+1 5284^172*172^5284+1 9218^184*184^9218+1 3772^187*187^3772+1 3688^188*188^3688+1 7492^190*190^7492+1 3281^192*192^3281+1 7836^197*197^7836+1 1/28/8 Blazek reports bases 2 and 3 checked to 100000, no new primes found. 5/21/7 John Blazek notes that base 2, primes 3 and 4 had been omitted. He's doubled checked base 2 to 50000 and also, checked base 3 to 50000. 1/2/7 Brown reports that bases 4000 to 4100 were checked through 20000. 10/27/6 Platzer reports base 5 range 16384 to 30000 checked, no primes; finds 4774^10*10^4774+1 is prime, with base 10 completed to 10000, reserves base 9 to 20000. 10/30/6 Platzer reports base 9 checked from 14001 to 20000, no primes found. send comments, corrections, additons, etc. to harvey563@yahoo.com 4:17 PM 6/19/2013