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

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

7111...1117 with 10907 digits: factordb stdkmd

666...66611 with 5039 digits: factordb stdkmd

95777...777 with 2906 digits: factordb stdkmd

999...99913 with 1800 digits: factordb stdkmd

45111...111 with 774 digits: factordb

111...11133 with 715 digits: factordb

40333...333 with 485 digits: factordb

4888...8881 with 464 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:

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

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

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

There is no forms aaa...aaabc, abbb...bbbc, abccc...cccc 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 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: 3, 3, 3

103: 3, 3, 3

107: 3, 3, 3

109: 3, 3, 3

111: 19, 19, 19

113: 3, 3, 3

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: 3, 3, 3

129: 4, 4, infinity (always divisible by 3)

131: 3, 3, 3

133: 715, 16, 16

137: 3, 3, 3

139: 3, 3, 3

141: 8, 7, 11

143: 6, infinity (always divisible by 13), 4

147: 32, 4, infinity (always divisible by 7)

149: 3, 3, 3

151: 3, 3, 3

153: 4, 4, infinity (always divisible by 3)

157: 3, 3, 3

159: 5, 4, infinity (always divisible by 3)

161: 5, 5, 5

163: 3, 3, 3

167: 3, 3, 3

169: 7, 4, 4

171: 4, 7, 10

173: 3, 3, 3

177: 5, 4, 4

179: 3, 3, 3

181: 3, 3, 3

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: 3, 3, 3

193: 3, 3, 3

197: 3, 3, 3

199: 3, 3, 3

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: 3, 3, 3

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: 3, 3, 3

227: 3, 3, 3

229: 3, 3, 3

231: 28, infinity (always divisible by 21), 4

233: 3, 3, 3

237: 4, infinity (always divisible by 3), 4

239: 3, 3, 3

241: 3, 3, 3

243: 4, 5, infinity (always divisible by 3)

247: 5, 4, 4

249: 7, 16, infinity (always divisible by 3)

251: 3, 3, 3

253: 8, infinity (always divisible by 23), 10

257: 3, 3, 3

259: 5, 83, 5

261: 8, infinity (always divisible by 3), 5

263: 3, 3, 3

267: 4, infinity (always divisible by 3), 4

269: 3, 3, 3

271: 3, 3, 3

273: 4, 5, infinity (always divisible by 3)

277: 3, 3, 3

279: 5, 5, infinity (always divisible by 9)

281: 3, 3, 3

283: 3, 3, 3

287: 4, 4, infinity (always divisible by 7)

289: 6, 8, 6

291: 5, infinity (always divisible by 3), 8

293: 3, 3, 3

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: 3, 3, 3

309: infinity (always divisible by 3), infinity (always divisible by 3), infinity (always divisible by 3)

311: 3, 3, 3

313: 3, 3, 3

317: 3, 3, 3

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: 3, 3, 3

333: infinity (always divisible by 3), infinity (always divisible by 3), infinity (always divisible by 3)

337: 3, 3, 3

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: 3, 3, 3

349: 3, 3, 3

351: infinity (always divisible by 3), 7, 4

353: 3, 3, 3

357: infinity (always divisible by 3), 4, infinity (always divisible by 7)

359: 3, 3, 3

361: 4, 7, 6

363: infinity (always divisible by 3), infinity (always divisible by 33), infinity (always divisible by 3)

367: 3, 3, 3

369: infinity (always divisible by 3), infinity (always divisible by 3), infinity (always divisible by 9)

371: 4, 6, infinity (see above)

373: 3, 3, 3

377: 5, 12, 12

379: 3, 3, 3

381: infinity (always divisible by 3), 4, infinity (see above)

383: 3, 3, 3

387: infinity (always divisible by 3), 7, 4

389: 3, 3, 3

391: 4, 18, 4

393: infinity (always divisible by 3), infinity (always divisible by 3), infinity (always divisible by 3)

397: 3, 3, 3

399: infinity (always divisible by 3), infinity (always divisible by 3), infinity (always divisible by 3)

401: 3, 3, 3

403: 6, 4, 485

407: infinity (see above), 4, 14

409: 3, 3, 3

411: 17, 4, 4

413: 11, 5, 4

417: 5, 5, 4

419: 3, 3, 3

421: 3, 3, 3

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: 3, 3, 3

433: 3, 3, 3

437: 9, 4, 5

439: 3, 3, 3

441: 4, 4, 5

443: 3, 3, 3

447: 4, 4, 5

449: 3, 3, 3

451: 4, infinity (always divisible by 41), 774

453: 5, 5, infinity (always divisible by 3)

457: 3, 3, 3

459: 8, 7, infinity (always divisible by 9)

461: 3, 3, 3

463: 3, 3, 3

467: 3, 3, 3

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: 3, 3, 3

481: 4, 464, 7

483: 4, 5, infinity (always divisible by 3)

487: 3, 3, 3

489: 8, 4, infinity (always divisible by 3)

491: 3, 3, 3

493: 4, 4, 4

497: 5, 16, infinity (always divisible by 7)

499: 3, 3, 3

501: 4, infinity (always divisible by 3), 4

503: 3, 3, 3

507: 4, infinity (always divisible by 3), 4

509: 3, 3, 3

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: 3, 3, 3

523: 3, 3, 3

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

541: 3, 3, 3

543: 8, 4, infinity (always divisible by 3)

547: 3, 3, 3

549: 70, 4, infinity (always divisible by 9)

551: 12, 12, 33

553: 8, 8, 5

557: 3, 3, 3

559: 8, 8, 7

561: 184, infinity (always divisible by 51), 18472

563: 3, 3, 3

567: 7, infinity (always divisible by 3), infinity (always divisible by 7)

569: 3, 3, 3

571: 3, 3, 3

573: 4, 5, infinity (always divisible by 3)

577: 3, 3, 3

579: 5, 4, infinity (always divisible by 3)

581: 4, 4, 5

583: 56, infinity (always divisible by 53), 8

587: 3, 3, 3

589: 5, 5, 17

591: 4, infinity (always divisible by 3), 20

593: 3, 3, 3

597: 26, infinity (always divisible by 3), 11

599: 3, 3, 3

601: 3, 3, 3

603: infinity (always divisible by 3), infinity (always divisible by 9), infinity (always divisible by 3)

607: 3, 3, 3

609: infinity (always divisible by 3), infinity (always divisible by 3), infinity (always divisible by 3)

611: 5039, 6, 6

613: 3, 3, 3

617: 3, 3, 3

619: 3, 3, 3

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: 3, 3, 3

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: 3, 3, 3

643: 3, 3, 3

647: 3, 3, 3

649: 6, 4, 14

651: infinity (always divisible by 3), 4, 5

653: 3, 3, 3

657: infinity (always divisible by 3), 5, 4

659: 3, 3, 3

661: 3, 3, 3

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: 3, 3, 3

677: 3, 3, 3

679: 4, 4, 6

681: infinity (always divisible by 3), 5, 5

683: 3, 3, 3

687: infinity (always divisible by 3), 34, 5

689: 4, 6, 4

691: 3, 3, 3

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: 3, 3, 3

703: 4, 5, 7

707: infinity (always divisible by 7), infinity (always divisible by 7), infinity (always divisible by 7)

709: 3, 3, 3

711: 5, 8, 8

713: 5, 8, 5

717: 4, 10907, 4

719: 3, 3, 3

721: infinity (always divisible by 7), 5, 4

723: 4, 5, infinity (always divisible by 3)

727: 3, 3, 3

729: 20, 4, infinity (always divisible by 9)

731: 5, 4, 9

733: 3, 3, 3

737: 6, infinity (always divisible by 11), 68

739: 3, 3, 3

741: 4, 5, 4

743: 3, 3, 3

747: 5, 11, 4

749: infinity (always divisible by 7), 5, 4

751: 3, 3, 3

753: 4, 5, infinity (always divisible by 3)

757: 3, 3, 3

759: 4, 4, infinity (always divisible by 3)

761: 3, 3, 3

763: infinity (always divisible by 7), 8, 5

767: 11, 5, 5

769: 3, 3, 3

771: 13, 13, 11

773: 3, 3, 3

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: 3, 3, 3

789: 4, 5, infinity (always divisible by 3)

791: infinity (always divisible by 7), 6, 5

793: 4, 4, 4

797: 3, 3, 3

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: 3, 3, 3

811: 3, 3, 3

813: 5, 11, infinity (always divisible by 3)

817: 5, 4, 6

819: 4, 5, infinity (always divisible by 9)

821: 3, 3, 3

823: 3, 3, 3

827: 3, 3, 3

829: 3, 3, 3

831: 4, infinity (always divisible by 3), 4

833: 79, 8, 8

837: 4, infinity (always divisible by 3), 4

839: 3, 3, 3

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: 3, 3, 3

857: 3, 3, 3

859: 3, 3, 3

861: 4, infinity (always divisible by 3), 5

863: 3, 3, 3

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: 3, 3, 3

879: 8, 4, infinity (always divisible by 3)

881: 3, 3, 3

883: 3, 3, 3

887: 3, 3, 3

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: 3, 3, 3

909: infinity (always divisible by 9), infinity (always divisible by 9), infinity (always divisible by 9)

911: 3, 3, 3

913: 1800, infinity (see above), 4

919: 3, 3, 3

921: infinity (always divisible by 3), 4, 5

923: 4, 6, 5

927: infinity (always divisible by 9), 4, 4

929: 3, 3, 3

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: 3, 3, 3

939: infinity (always divisible by 3), infinity (always divisible by 3), infinity (always divisible by 3)

941: 3, 3, 3

943: 7, 31, 4

947: 3, 3, 3

949: 4, infinity (see above), 5

951: infinity (always divisible by 3), 4, 4

953: 3, 3, 3

957: infinity (always divisible by 3), 8, 2906

959: 6, infinity (see above), 12

963: infinity (always divisible by 9), infinity (always divisible by 3), infinity (always divisible by 3)

967: 3, 3, 3

969: infinity (always divisible by 3), infinity (always divisible by 3), infinity (always divisible by 3)

971: 3, 3, 3

973: 4, 7, 4

977: 3, 3, 3

979: 6, infinity (always divisible by 11), 92

981: infinity (always divisible by 9), 7, 4

983: 3, 3, 3

987: infinity (always divisible by 3), 4, infinity (always divisible by 7)

989: 5, 7, 5

991: 3, 3, 3

993: infinity (always divisible by 3), infinity (always divisible by 3), infinity (always divisible by 3)

997: 3, 3, 3

999: infinity (always divisible by 9), infinity (always divisible by 9), infinity (always divisible by 9)