Format:
Three-digit number coprime to 10 (abc), length of the smallest prime of the form aaa...aaabc, length of the smallest prime of the form abbb...bbbc, length of the smallest prime of the form abccc...cccc
Note:
The three-digit number abc itself and the two-digit numbers ab, ac, bc are not counted
Links to the factordb entries of the primes > 10^299 and stdkmd entries of the primes > 10^999, with PRIMO primality certificates of them, to prove that they are actually primes and not merely PRPs, these large primes are: (primes < 10^299 can be verified in a few seconds)
222...22299 with 19153 digits: factordb stdkmd
56111...111 with 18472 digits: factordb stdkmd
44999...999 with 11960 digits: (proven prime by N+1 method and N+1 is trivially fully factored)
7111...1117 with 10907 digits: factordb stdkmd
666...66611 with 5039 digits: factordb stdkmd
95777...777 with 2906 digits: factordb stdkmd
33777...777 with 2187 digits: factordb stdkmd
999...99913 with 1800 digits: factordb stdkmd
45111...111 with 774 digits: factordb
3444...4447 with 759 digits: factordb
111...11133 with 715 digits: factordb
999...99911 with 637 digits: (proven prime by N-1 method and N-1 has difference-of-two-squares factorization, and the largest factor of an algebraic factor of N-1 has only 268 digits)
222...22211 with 495 digits: (proven prime by N-1 method and the certificate of the large prime factor of N-1)
40333...333 with 485 digits: factordb
4888...8881 with 464 digits: factordb
222...22233 with 303 digits: factordb
Some forms contain no primes, if abc is not coprime to (aabc, abbc, abcc), then (respectively) (aaa...aaabc, abbb...bbbc, abccc...cccc) contain no primes, since all numbers in these forms are divisible by the gcd, but there are also other cases: (reference: https://stdkmd.net/nrr/coveringset.htm and http://irvinemclean.com/maths/siercvr.htm)
37111...111:
divisible by 3 if the number of 1's is == 2 mod 3
divisible by 37 if the number of 1's is == 0 mod 3
divisible by 7 if the number of 1's is == 1 mod 6
divisible by 13 if the number of 1's is == 4 mod 6
38111...111:
divisible by 3 if the number of 1's is == 1 mod 3
divisible by 37 if the number of 1's is == 2 mod 3
divisible by 2333...333 if the number of 1's is == 0 mod 3 and the number of 3's in this factor is 1/3 of the number of 1's (difference-of-two-cubes factorization)
444...44407:
divisible by 11 if the number of 4's is == 1 mod 2
divisible by 3 if the number of 4's is == 2 mod 3
divisible by 37 if the number of 4's is == 1 mod 3
divisible by 7 if the number of 4's is == 0 mod 6
444...44407 also:
divisible by 11 if the number of 4's is == 1 mod 2
divisible by 37 if the number of 4's is == 1 mod 3
divisible by 7 if the number of 4's is == 0 mod 6
divisible by 13 if the number of 4's is == 2 mod 6
555...55537:
divisible by 3 if the number of 5's is == 1 mod 3
divisible by 37 if the number of 5's is == 0 mod 3
divisible by 7 if the number of 5's is == 2 mod 6
divisible by 13 if the number of 5's is == 5 mod 6
9111...1113:
divisible by 11 if the number of 1's is == 1 mod 2
divisible by 3 if the number of 1's is == 0 mod 3
divisible by 7 if the number of 1's is == 4 mod 6
divisible by 13 if the number of 1's is == 2 mod 6
9444...4449:
divisible by 11 if the number of 4's is == 0 mod 2
divisible by 3 if the number of 4's is == 0 mod 3
divisible by 7 if the number of 4's is == 5 mod 6
divisible by 13 if the number of 4's is == 1 mod 6
9555...5559:
divisible by 11 if the number of 5's is == 0 mod 2
divisible by 3 if the number of 5's is == 0 mod 3
divisible by 7 if the number of 5's is == 1 mod 6
divisible by 13 if the number of 5's is == 5 mod 6
Also the family 1000...0001: If the number 1000...0001 is prime, then the number of 0's must be of the form 2^n-1 (since it is divisible by 11 if the number of 0's is == 0 mod 2, divisible by 101 if the number of 0's is == 1 mod 4, divisible by 10001 if the number of 0's is == 3 mod 8, divisible by 100000001 if the number of 0's is == 7 mod 16, etc.), and if such prime exists, then there must be at least 2^31-1 0's, see http://www.prothsearch.com/GFN10.html
There are ZERO forms aaa...aaabc, abbb...bbbc, abccc...cccc other than 1000...0001 with no known primes neither can be proven to contain no primes, and all smallest primes of the forms are proven to be primes (i.e. not merely PRPs), thus now there is a THEOREM:
Forms aaa...aaabc, abbb...bbbc, abccc...cccc other than 1000...0001 contain a prime if and only if aaa...aaabc, abbb...bbbc, abccc...cccc is not divisible by a single prime factor and aaa...aaabc, abbb...bbbc, abccc...cccc is not the seven forms above (i.e. 37111...111, 38111...111, 444...44407, 555...55537, 9111...1113, 9444...4449, 9555...5559)
Data:
101: 8, >=2147483649 (see above), 5
103: 4, 6, 4
107: 17, 5, 6
109: 4, 4, 11
111: 19, 19, 19
113: 5, 5, 7
117: 4, 4, 5
119: 5, 5, 7
121: 6, infinity (always divisible by 11), 138
123: 4, 4, infinity (always divisible by 3)
127: 6, 5, 4
129: 4, 4, infinity (always divisible by 3)
131: 5, 5, 6
133: 715, 16, 16
137: 117, 5, 6
139: 9, 5, 4
141: 8, 7, 11
143: 6, infinity (always divisible by 13), 4
147: 32, 4, infinity (always divisible by 7)
149: 5, 5, 4
151: 4, 5, 4
153: 4, 4, infinity (always divisible by 3)
157: 7, 6, 10
159: 5, 4, infinity (always divisible by 3)
161: 5, 5, 5
163: 4, 4, 5
167: 51, 4, 6
169: 7, 4, 4
171: 4, 7, 10
173: 5, 9, 4
177: 5, 4, 4
179: 21, 9, 6
181: 4, 9, 4
183: 7, 17, infinity (always divisible by 3)
187: 4, infinity (always divisible by 17), 4
189: 7, 4, infinity (always divisible by 9)
191: 6, 5, 14
193: 4, 4, 4
197: 5, 4, 5
199: 9, 4, 4
201: 14, infinity (always divisible by 3), 4
203: 4, 4, 5
207: 4, infinity (always divisible by 9), 8
209: 28, 6, 4
211: 495, 4, 4
213: 4, 4, infinity (always divisible by 3)
217: 206, 37, infinity (always divisible by 7)
219: 7, 224, infinity (always divisible by 3)
221: 4, 4, 5
223: 8, 8, 9
227: 9, 9, 5
229: 5, 5, 63
231: 28, infinity (always divisible by 21), 4
233: 303, 4, 4
237: 4, infinity (always divisible by 3), 4
239: 4, 4, 4
241: 9, 4, 4
243: 4, 5, infinity (always divisible by 3)
247: 5, 4, 4
249: 7, 16, infinity (always divisible by 3)
251: 4, 4, 5
253: 8, infinity (always divisible by 23), 10
257: 33, 4, 20
259: 5, 83, 5
261: 8, infinity (always divisible by 3), 5
263: 7, 4, 4
267: 4, infinity (always divisible by 3), 4
269: 4, 5, 4
271: 5, 7, 4
273: 4, 5, infinity (always divisible by 3)
277: 5, 4, 4
279: 5, 5, infinity (always divisible by 9)
281: 4, 13, 5
283: 5, 8, 4
287: 4, 4, infinity (always divisible by 7)
289: 6, 8, 6
291: 5, infinity (always divisible by 3), 8
293: 4, 6, 5
297: 4, infinity (always divisible by 27), 50
299: 19153, 4, 4
301: 4, 4, 4
303: infinity (always divisible by 3), infinity (always divisible by 3), infinity (always divisible by 3)
307: 4, 6, 9
309: infinity (always divisible by 3), infinity (always divisible by 3), infinity (always divisible by 3)
311: 5, 6, 6
313: 4, 13, 5
317: 5, 219, 6
319: 4, 4, 30
321: infinity (always divisible by 3), 4, 37
323: 4, 7, 6
327: infinity (always divisible by 3), 43, 7
329: 4, 4, 4
331: 4, 4, 27
333: infinity (always divisible by 3), infinity (always divisible by 3), infinity (always divisible by 3)
337: 6, 6, 2187
339: infinity (always divisible by 3), infinity (always divisible by 3), infinity (always divisible by 3)
341: 6, infinity (always divisible by 31), 8
343: 4, 7, 4
347: 4, 759, 9
349: 5, 4, 4
351: infinity (always divisible by 3), 7, 4
353: 5, 9, 4
357: infinity (always divisible by 3), 4, infinity (always divisible by 7)
359: 4, 4, 5
361: 4, 7, 6
363: infinity (always divisible by 3), infinity (always divisible by 33), infinity (always divisible by 3)
367: 6, 13, 4
369: infinity (always divisible by 3), infinity (always divisible by 3), infinity (always divisible by 9)
371: 4, 6, infinity (see above)
373: 4, 15, 4
377: 5, 12, 12
379: 9, 4, 6
381: infinity (always divisible by 3), 4, infinity (see above)
383: 6, 13, 4
387: infinity (always divisible by 3), 7, 4
389: 4, 4, 6
391: 4, 18, 4
393: infinity (always divisible by 3), infinity (always divisible by 3), infinity (always divisible by 3)
397: 6, 15, 7
399: infinity (always divisible by 3), infinity (always divisible by 3), infinity (always divisible by 3)
401: 6, 4, 5
403: 6, 4, 485
407: infinity (see above), 4, 14
409: 4, 5, 4
411: 17, 4, 4
413: 11, 5, 4
417: 5, 5, 4
419: 18, 6, 5
421: 4, 5, 4
423: 4, 5, infinity (always divisible by 3)
427: 52, 5, infinity (always divisible by 7)
429: 10, 4, infinity (always divisible by 3)
431: 53, 5, 9
433: 51, 17, 17
437: 9, 4, 5
439: 13, 4, 8
441: 4, 4, 5
443: 6, 6, 7
447: 4, 4, 5
449: 5, 5, 11960
451: 4, infinity (always divisible by 41), 774
453: 5, 5, infinity (always divisible by 3)
457: 4, 5, 18
459: 8, 7, infinity (always divisible by 9)
461: 6, 8, 20
463: 4, 4, 8
467: 8, 7, 51
469: 6, infinity (always divisible by 7), 18
471: 7, 23, 5
473: 6, infinity (always divisible by 43), 4
477: 17, 5, 5
479: 8, 5, 4
481: 4, 464, 7
483: 4, 5, infinity (always divisible by 3)
487: 6, 8, 4
489: 8, 4, infinity (always divisible by 3)
491: 5, 5, 12
493: 4, 4, 4
497: 5, 16, infinity (always divisible by 7)
499: 21, 4, 4
501: 4, infinity (always divisible by 3), 4
503: 4, 4, 5
507: 4, infinity (always divisible by 3), 4
509: 7, 4, 4
511: 5, 6, 6
513: 71, 4, infinity (always divisible by 3)
517: 8, 24, 10
519: 4, 4, infinity (always divisible by 3)
521: 4, 10, 47
523: 6, 5, 4
527: 4, 4, 9
529: 5, 6, 5
531: 4, infinity (always divisible by 3), 11
533: 19, 4, 4
537: infinity (see above), infinity (always divisible by 3), 5
539: 16, 48, 4
541: 5, 4, 9
543: 8, 4, infinity (always divisible by 3)
547: 5, 7, 4
549: 70, 4, infinity (always divisible by 9)
551: 12, 12, 33
553: 8, 8, 5
557: 4, 4, 185
559: 8, 8, 7
561: 184, infinity (always divisible by 51), 18472
563: 4, 5, 5
567: 7, infinity (always divisible by 3), infinity (always divisible by 7)
569: 4, 4, 5
571: 45, 7, 4
573: 4, 5, infinity (always divisible by 3)
577: 51, 9, 9
579: 5, 4, infinity (always divisible by 3)
581: 4, 4, 5
583: 56, infinity (always divisible by 53), 8
587: 9, 7, 9
589: 5, 5, 17
591: 4, infinity (always divisible by 3), 20
593: 6, 6, 5
597: 26, infinity (always divisible by 3), 11
599: 9, 5, 5
601: 5, 9, 4
603: infinity (always divisible by 3), infinity (always divisible by 9), infinity (always divisible by 3)
607: 4, 4, 10
609: infinity (always divisible by 3), infinity (always divisible by 3), infinity (always divisible by 3)
611: 5039, 6, 6
613: 7, 4, 4
617: 5, 33, 6
619: 4, 261, 4
621: infinity (always divisible by 3), 4, 4
623: 32, infinity (always divisible by 7), 7
627: infinity (always divisible by 3), 14, 4
629: 5, 4, 4
631: 8, 5, 4
633: infinity (always divisible by 3), infinity (always divisible by 3), infinity (always divisible by 3)
637: 4, 4, infinity (always divisible by 7)
639: infinity (always divisible by 3), infinity (always divisible by 3), infinity (always divisible by 9)
641: 14, 8, 14
643: 5, 6, 5
647: 6, 6, 8
649: 6, 4, 14
651: infinity (always divisible by 3), 4, 5
653: 4, 4, 53
657: infinity (always divisible by 3), 5, 4
659: 4, 6, 4
661: 4, 4, 195
663: infinity (always divisible by 3), infinity (always divisible by 3), infinity (always divisible by 3)
667: 6, 6, 7
669: infinity (always divisible by 3), infinity (always divisible by 3), infinity (always divisible by 3)
671: 6, infinity (always divisible by 61), 212
673: 4, 157, 4
677: 19, 5, 5
679: 4, 4, 6
681: infinity (always divisible by 3), 5, 5
683: 5, 4, 4
687: infinity (always divisible by 3), 34, 5
689: 4, 6, 4
691: 4, 4, 4
693: infinity (always divisible by 3), infinity (always divisible by 63), infinity (always divisible by 3)
697: 5, 4, 4
699: infinity (always divisible by 3), infinity (always divisible by 3), infinity (always divisible by 3)
701: 9, 4, 5
703: 4, 5, 7
707: infinity (always divisible by 7), infinity (always divisible by 7), infinity (always divisible by 7)
709: 7, 5, 5
711: 5, 8, 8
713: 5, 8, 5
717: 4, 10907, 4
719: 5, 5, 5
721: infinity (always divisible by 7), 5, 4
723: 4, 5, infinity (always divisible by 3)
727: 4, 5, 6
729: 20, 4, infinity (always divisible by 9)
731: 5, 4, 9
733: 51, 4, 4
737: 6, infinity (always divisible by 11), 68
739: 9, 6, 5
741: 4, 5, 4
743: 5, 20, 4
747: 5, 11, 4
749: infinity (always divisible by 7), 5, 4
751: 9, 6, 9
753: 4, 5, infinity (always divisible by 3)
757: 4, 5, 4
759: 4, 4, infinity (always divisible by 3)
761: 5, 7, 14
763: infinity (always divisible by 7), 8, 5
767: 11, 5, 5
769: 6, 4, 4
771: 13, 13, 11
773: 5, 5, 9
777: infinity (always divisible by 7), infinity (always divisible by 7), infinity (always divisible by 7)
779: 66, 66, 5
781: 6, infinity (always divisible by 71), 6
783: 5, 4, infinity (always divisible by 3)
787: 6, 5, 4
789: 4, 5, infinity (always divisible by 3)
791: infinity (always divisible by 7), 6, 5
793: 4, 4, 4
797: 5, 5, 5
799: 9, 5, 5
801: 5, infinity (always divisible by 9), 4
803: 4, 32, 6
807: 4, infinity (always divisible by 3), 5
809: 6, 4, 7
811: 5, 4, 4
813: 5, 11, infinity (always divisible by 3)
817: 5, 4, 6
819: 4, 5, infinity (always divisible by 9)
821: 4, 4, 14
823: 26, 5, 4
827: 6, 21, 9
829: 9, 6, 13
831: 4, infinity (always divisible by 3), 4
833: 79, 8, 8
837: 4, infinity (always divisible by 3), 4
839: 4, 5, 6
841: 8, 27, 7
843: 5, 4, infinity (always divisible by 3)
847: 32, 4, infinity (always divisible by 7)
849: 4, 5, infinity (always divisible by 3)
851: 7, 22, 8
853: 5, 8, 5
857: 6, 9, 8
859: 9, 9, 4
861: 4, infinity (always divisible by 3), 5
863: 4, 4, 10
867: 4, infinity (always divisible by 3), 4
869: 6, 4, 4
871: 6, 6, 151
873: 5, 8, infinity (always divisible by 3)
877: 27, 10, 10
879: 8, 4, infinity (always divisible by 3)
881: 19, 19, 15
883: 5, 5, 255
887: 4, 4, 15
889: 14, 14, 35
891: 10, infinity (always divisible by 81), 8
893: 4, 7, 4
897: 5, infinity (always divisible by 3), 10
899: 45, 4, 4
901: 4, 4, 4
903: infinity (always divisible by 3), infinity (always divisible by 3), infinity (always divisible by 3)
907: 4, 4, 7
909: infinity (always divisible by 9), infinity (always divisible by 9), infinity (always divisible by 9)
911: 637, 6, 6
913: 1800, infinity (see above), 4
917: 6, 18, infinity (always divisible by 7)
919: 17, 247, 4
921: infinity (always divisible by 3), 4, 5
923: 4, 6, 5
927: infinity (always divisible by 9), 4, 4
929: 4, 7, 10
931: 4, infinity (always divisible by 7), 4
933: infinity (always divisible by 3), infinity (always divisible by 3), infinity (always divisible by 3)
937: 7, 4, 4
939: infinity (always divisible by 3), infinity (always divisible by 3), infinity (always divisible by 3)
941: 4, 5, 5
943: 7, 31, 4
947: 11, 5, 5
949: 4, infinity (see above), 5
951: infinity (always divisible by 3), 4, 4
953: 6, 21, 4
957: infinity (always divisible by 3), 8, 2906
959: 6, infinity (see above), 12
961: 5, 4, 9
963: infinity (always divisible by 9), infinity (always divisible by 3), infinity (always divisible by 3)
967: 4, 5, 4
969: infinity (always divisible by 3), infinity (always divisible by 3), infinity (always divisible by 3)
971: 5, 5, 6
973: 4, 7, 4
977: 11, 5, 5
979: 6, infinity (always divisible by 11), 92
981: infinity (always divisible by 9), 7, 4
983: 6, 4, 4
987: infinity (always divisible by 3), 4, infinity (always divisible by 7)
989: 5, 7, 5
991: 5, 5, 9
993: infinity (always divisible by 3), infinity (always divisible by 3), infinity (always divisible by 3)
997: 17, 17, 7
999: infinity (always divisible by 9), infinity (always divisible by 9), infinity (always divisible by 9)