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Primi cancellabili

Rappresentazione dei numeri 

Si chiamano “cancellabili” i numeri primi, tali che, cancellando una alla volta le cifre in qualche ordine, rimangono primi sino a ridursi a una sola cifra. Comprendono quindi i primi troncabili come caso particolare.

Per esempio, è cancellabile 9871, perché si possono cancellare le cifre una alla volta, ottenendo i primi 971, 97, 7.

 

I primi cancellabili minori di 1000 sono: 2, 3, 5, 7, 13, 17, 23, 29, 31, 37, 43, 47, 53, 59, 67, 71, 73, 79, 83, 97, 103, 107, 113, 127, 131, 137, 139, 157, 163, 167, 173, 179, 193, 197, 223, 229, 233, 239, 263, 269, 271, 283, 293, 307, 311, 313, 317, 331, 337, 347, 353, 359, 367, 373, 379, 383, 397, 431, 433, 439, 443, 457, 463, 467, 479, 487, 503, 509, 523, 547, 563, 569, 571, 593, 599, 607, 613, 617, 631, 643, 647, 653, 659, 673, 677, 683, 701, 709, 719, 733, 739, 743, 751, 761, 769, 773, 797, 823, 829, 839, 853, 859, 863, 883, 907, 929, 937, 947, 953, 967, 971, 977, 983, 997.

In questo caso si ammette l’uso dello zero, che va cancellato come le altre cifre, ma non deve rimanere come prima cifra a sinistra.

 

Si suppone che sia i primi cancellabili, sia quelli non cancellabili siano infiniti, però nessuna delle due affermazioni è stata dimostrata.

Nel 2003 K. Andersen dmostrò che se in un primo la differenza tra il numero di cifre dell’insieme { 1, 4, 7 } e quello delle cifre dell’insieme {2, 5, 8 } è maggiore di 2, il primo non è cancellabile.

 

Tra i maggiori noti ne segnalo uno di ben 300 cifre: 7219606290619041107569730478968205296506466269139879141930580819468693003235056315036137851472350837475985290920679625126053388015816842349644908050702945726184979336539489066941340113614760956506598624163760290701029109701289909955048647280564362349 36733309062195718092793663000312918831684212336353.

 

Il massimo primo cancellabile noto è probabilmente 60406046028060460860686400084886624406680640804422620006048062042680626006800684446808646204066660646280422486804668040804640804626406002408068642060006402460406266206604404060460006246468440662462828006626804646046604080460064462066648406660044080606808044602646068066202846084044066204020408040406626042228060684084604262004466046066848460460028060666064604820260802400684004408604068400620440402040620420000484846606660206086040684660068600806684668600846084240480464046064004646260460086460260423 (Paul Jobling 2001, 500 cifre); scrivo “probabilmente” perché molti primi coinvolti nella sua costruzione sono primi probabili.

 

La definizione può essere estesa ad altre basi. In base 2 si considera primo 1, altrimenti nessun primo sarebbe cancellabile; per esempio, si considerano cancellabili 3 = 112 e 5 = 1012, anche se dopo l’ultima cancellazione si resta con 1.

 

La tabella seguente riporta i primi cancellabili minori di 1000 nelle basi da 2 a 20.

Base

Primi

2

112 = 3, 1012 = 5, 1112 = 7, 10112 = 11, 11012 = 13, 100112 = 19, 101112 = 23, 111012 = 29, 1001012 = 37, 1010112 = 43, 1011112 = 47, 1101012 = 53, 1110112 = 59, 1111012 = 61, 10010012 = 73, 10011112 = 79, 10100112 = 83, 11001012 = 101, 11010112 = 107, 11011012 = 109, 100010012 = 137, 100101012 = 149, 100101112 = 151, 100111012 = 157, 101000112 = 163, 101001112 = 167, 101011012 = 173, 101100112 = 179, 101101012 = 181, 110001012 = 197, 110100112 = 211, 111001012 = 229, 1000101012 = 277, 1000110012 = 281, 1001001012 = 293, 1001100112 = 307, 1001101112 = 311, 1001110012 = 313, 1001111012 = 317, 1010010112 = 331, 1010110112 = 347, 1010111012 = 349, 1011001112 = 359, 1011101012 = 373, 1100001012 = 389, 1100011012 = 397, 1101000112 = 419, 1101001012 = 421, 1110010012 = 457, 1110011012 = 461, 1110100112 = 467, 10001011012 = 557, 10001100112 = 563, 10001110012 = 569, 10010010112 = 587, 10010101112 = 599, 10010110012 = 601, 10011001012 = 613, 10011010012 = 617, 10011010112 = 619, 10011101112 = 631, 10100011012 = 653, 10100100112 = 659, 10100101012 = 661, 10101001012 = 677, 10101010112 = 683, 10101100112 = 691, 10101111012 = 701, 10110001012 = 709, 10110011112 = 719, 10110101112 = 727, 10110111012 = 733, 10111001112 = 743, 10111101012 = 757, 11000001012 = 773, 11000111012 = 797, 11001101112 = 823, 11001111012 = 829, 11010001112 = 839, 11010101012 = 853, 11010110112 = 859, 11101010012 = 937, 11101011012 = 941, 11101100112 = 947

3

23 = 2, 123 = 5, 213 = 7, 1023 = 11, 1223 = 17, 2013 = 19, 2123 = 23, 10023 = 29, 12023 = 47, 12223 = 53, 20123 = 59, 20213 = 61, 21223 = 71, 22013 = 73, 100023 = 83, 100223 = 89, 102023 = 101, 102223 = 107, 120023 = 137, 200123 = 167, 201023 = 173, 201223 = 179, 202013 = 181, 210023 = 191, 210223 = 197, 220213 = 223, 221223 = 233, 1000223 = 251, 1002023 = 263, 1002223 = 269, 1022023 = 317, 1202223 = 431, 1220023 = 461, 2000123 = 491, 2001223 = 503, 2002123 = 509, 2010223 = 521, 2020013 = 541, 2020213 = 547, 2021223 = 557, 2100023 = 569, 2102023 = 587, 2102223 = 593, 2200123 = 653, 2201023 = 659, 2202213 = 673, 2210023 = 677, 2210223 = 683, 2212223 = 701, 2220213 = 709, 2221223 = 719, 10022223 = 809, 10202023 = 911, 10220023 = 947, 10220223 = 953

4

24 = 2, 34 = 3, 134 = 7, 234 = 11, 314 = 13, 1034 = 19, 1134 = 23, 1314 = 29, 1334 = 31, 2234 = 43, 2334 = 47, 3114 = 53, 3234 = 59, 3314 = 61, 10034 = 67, 10134 = 71, 10334 = 79, 11034 = 83, 12134 = 103, 12234 = 107, 12314 = 109, 13014 = 113, 13334 = 127, 20234 = 139, 21134 = 151, 21314 = 157, 22034 = 163, 22134 = 167, 22314 = 173, 23034 = 179, 23114 = 181, 23334 = 191, 30114 = 197, 30134 = 199, 31034 = 211, 31334 = 223, 32034 = 227, 32114 = 229, 32334 = 239, 33014 = 241, 33234 = 251, 100134 = 263, 100314 = 269, 100334 = 271, 101234 = 283, 103034 = 307, 103134 = 311, 103314 = 317, 110234 = 331, 112134 = 359, 113334 = 383, 120314 = 397, 122034 = 419, 122334 = 431, 123014 = 433, 123134 = 439, 123234 = 443, 130014 = 449, 130214 = 457, 130314 = 461, 130334 = 463, 131034 = 467, 131334 = 479, 132134 = 487, 132234 = 491, 133034 = 499, 133134 = 503, 133314 = 509, 200234 = 523, 201314 = 541, 202034 = 547, 202314 = 557, 203034 = 563, 203234 = 571, 210234 = 587, 211134 = 599, 211334 = 607, 212234 = 619, 213134 = 631, 220034 = 643, 220134 = 647, 220314 = 653, 221034 = 659, 223034 = 691, 223314 = 701, 230114 = 709, 230334 = 719, 231134 = 727, 231314 = 733, 232034 = 739, 232134 = 743, 232334 = 751, 233114 = 757, 300114 = 773, 301034 = 787, 301314 = 797, 302234 = 811, 303114 = 821, 303134 = 823, 303234 = 827, 303314 = 829, 310134 = 839, 311334 = 863, 312314 = 877, 313014 = 881, 313034 = 883, 313134 = 887, 320234 = 907, 320334 = 911, 321134 = 919, 322314 = 941, 323034 = 947, 330134 = 967, 330234 = 971, 331014 = 977, 331334 = 991, 332114 = 997

5

25 = 2, 35 = 3, 125 = 7, 215 = 11, 235 = 13, 325 = 17, 345 = 19, 435 = 23, 1225 = 37, 1425 = 47, 2035 = 53, 2145 = 59, 2215 = 61, 2325 = 67, 2415 = 71, 2435 = 73, 3045 = 79, 3245 = 89, 3425 = 97, 4035 = 103, 4125 = 107, 4235 = 113, 10225 = 137, 12425 = 197, 14025 = 227, 14245 = 239, 20125 = 257, 20235 = 263, 20345 = 269, 20415 = 271, 22125 = 307, 22215 = 311, 22325 = 317, 23225 = 337, 23425 = 347, 24035 = 353, 24145 = 359, 24325 = 367, 24435 = 373, 30045 = 379, 30245 = 389, 30425 = 397, 32245 = 439, 32445 = 449, 34045 = 479, 34225 = 487, 40035 = 503, 40435 = 523, 41425 = 547, 42125 = 557, 42235 = 563, 42345 = 569, 42415 = 571, 44125 = 607, 44235 = 613, 102025 = 677, 104025 = 727, 104245 = 739, 120225 = 887, 122425 = 947, 124025 = 977, 124425 = 997

6

26 = 2, 36 = 3, 56 = 5, 156 = 11, 216 = 13, 256 = 17, 316 = 19, 356 = 23, 456 = 29, 516 = 31, 1056 = 41, 1156 = 47, 1256 = 53, 1356 = 59, 1516 = 67, 1556 = 71, 2016 = 73, 2116 = 79, 2156 = 83, 2256 = 89, 2416 = 97, 2456 = 101, 2516 = 103, 2556 = 107, 3016 = 109, 3056 = 113, 3316 = 127, 3356 = 131, 3456 = 137, 3516 = 139, 4056 = 149, 4216 = 157, 4316 = 163, 4356 = 167, 4456 = 173, 4556 = 179, 5016 = 181, 5156 = 191, 5216 = 193, 5256 = 197, 5316 = 199, 5516 = 211, 10156 = 227, 10216 = 229, 10256 = 233, 10356 = 239, 10556 = 251, 11056 = 257, 11156 = 263, 11256 = 269, 11456 = 281, 11516 = 283, 12056 = 293, 12356 = 311, 12416 = 313, 12456 = 317, 13356 = 347, 13456 = 353, 13556 = 359, 14216 = 373, 14316 = 379, 14356 = 383, 14456 = 389, 15016 = 397, 15056 = 401, 15216 = 409, 15356 = 419, 15416 = 421, 15556 = 431, 20016 = 433, 20116 = 439, 20156 = 443, 20256 = 449, 20416 = 457, 20456 = 461, 20516 = 463, 20556 = 467, 21156 = 479, 21316 = 487, 21356 = 491, 21516 = 499, 21556 = 503, 22056 = 509, 22256 = 521, 23016 = 541, 23116 = 547, 23256 = 557, 23356 = 563, 23456 = 569, 23516 = 571, 24016 = 577, 24156 = 587, 24256 = 593, 24356 = 599, 24416 = 601, 24516 = 607, 25016 = 613, 25056 = 617, 25116 = 619, 25316 = 631, 25456 = 641, 25516 = 643, 25556 = 647, 30056 = 653, 30156 = 659, 30216 = 661, 30416 = 673, 30456 = 677, 30556 = 683, 31256 = 701, 31556 = 719, 32116 = 727, 32316 = 739, 32356 = 743, 32516 = 751, 33016 = 757, 33056 = 761, 33216 = 769, 33256 = 773, 33516 = 787, 34056 = 797, 34256 = 809, 34316 = 811, 34456 = 821, 34516 = 823, 34556 = 827, 35016 = 829, 35156 = 839, 35416 = 853, 35456 = 857, 35516 = 859, 40216 = 877, 40256 = 881, 40316 = 883, 40356 = 887, 41156 = 911, 41316 = 919, 41456 = 929, 42016 = 937, 42056 = 941, 42156 = 947, 42256 = 953, 42516 = 967, 42556 = 971, 43056 = 977, 43156 = 983, 43316 = 991, 43416 = 997

7

27 = 2, 37 = 3, 57 = 5, 237 = 17, 257 = 19, 327 = 23, 437 = 31, 527 = 37, 567 = 41, 657 = 47, 2037 = 101, 2057 = 103, 2457 = 131, 2547 = 137, 2567 = 139, 3027 = 149, 3227 = 163, 3267 = 167, 3627 = 191, 4037 = 199, 4367 = 223, 4437 = 227, 4527 = 233, 4637 = 241, 5067 = 251, 5247 = 263, 5667 = 293, 6237 = 311, 6257 = 313, 6327 = 317, 6527 = 331, 20057 = 691, 20327 = 709, 20457 = 719, 20567 = 727, 20657 = 733, 22037 = 787, 22547 = 823, 23267 = 853, 23627 = 877, 24057 = 887, 24527 = 919, 24657 = 929, 25067 = 937, 25647 = 977, 26037 = 983, 26237 = 997

8

28 = 2, 38 = 3, 58 = 5, 78 = 7, 138 = 11, 158 = 13, 218 = 17, 238 = 19, 278 = 23, 358 = 29, 378 = 31, 458 = 37, 518 = 41, 538 = 43, 578 = 47, 658 = 53, 738 = 59, 758 = 61, 1038 = 67, 1078 = 71, 1238 = 83, 1318 = 89, 1458 = 101, 1538 = 107, 1558 = 109, 2038 = 131, 2118 = 137, 2138 = 139, 2278 = 151, 2358 = 157, 2438 = 163, 2478 = 167, 2638 = 179, 2658 = 181, 2778 = 191, 3058 = 197, 3078 = 199, 3238 = 211, 3378 = 223, 3458 = 229, 3518 = 233, 3578 = 239, 3738 = 251, 4078 = 263, 4158 = 269, 4258 = 277, 4458 = 293, 4758 = 317, 5138 = 331, 5218 = 337, 5338 = 347, 5358 = 349, 5418 = 353, 5478 = 359, 5578 = 367, 5658 = 373, 5738 = 379, 5778 = 383, 6058 = 389, 6158 = 397, 6218 = 401, 6458 = 421, 6578 = 431, 6738 = 443, 7158 = 461, 7238 = 467, 7378 = 479, 7538 = 491, 7638 = 499, 7758 = 509, 10138 = 523, 10358 = 541, 10438 = 547, 10558 = 557, 10638 = 563, 10718 = 569, 10738 = 571, 11318 = 601, 11458 = 613, 11538 = 619, 12038 = 643, 12078 = 647, 12238 = 659, 12458 = 677, 12538 = 683, 12638 = 691, 13058 = 709, 13178 = 719, 13578 = 751, 13718 = 761, 14058 = 773, 14238 = 787, 14358 = 797, 14518 = 809, 14538 = 811, 14658 = 821, 14758 = 829, 15078 = 839, 15258 = 853, 15318 = 857, 15338 = 859, 15378 = 863, 15558 = 877, 15638 = 883, 16558 = 941, 17078 = 967, 17378 = 991, 17458 = 997

9

29 = 2, 39 = 3, 59 = 5, 79 = 7, 129 = 11, 219 = 19, 259 = 23, 329 = 29, 349 = 31, 459 = 41, 479 = 43, 529 = 47, 589 = 53, 659 = 59, 679 = 61, 749 = 67, 789 = 71, 879 = 79, 1029 = 83, 1229 = 101, 1249 = 103, 1289 = 107, 1629 = 137, 2019 = 163, 2059 = 167, 2129 = 173, 2189 = 179, 2219 = 181, 2329 = 191, 2349 = 193, 2419 = 199, 2549 = 211, 2679 = 223, 2749 = 229, 2789 = 233, 2859 = 239, 2879 = 241, 3229 = 263, 3289 = 269, 3429 = 281, 3449 = 283, 3829 = 317, 4079 = 331, 4259 = 347, 4279 = 349, 4329 = 353, 4479 = 367, 4549 = 373, 4659 = 383, 4729 = 389, 4859 = 401, 5289 = 431, 5429 = 443, 5489 = 449, 5629 = 461, 5689 = 467, 5829 = 479, 6059 = 491, 6259 = 509, 6679 = 547, 6789 = 557, 6859 = 563, 7029 = 569, 7049 = 571, 7289 = 593, 7449 = 607, 7829 = 641, 7849 = 643, 7889 = 647, 8059 = 653, 8129 = 659, 8279 = 673, 8329 = 677, 8479 = 691, 8589 = 701, 8679 = 709, 8789 = 719, 8879 = 727, 10249 = 751, 12229 = 911, 12429 = 929, 12629 = 947, 12689 = 953, 12849 = 967, 12889 = 971

10

2, 3, 5, 7, 13, 17, 23, 29, 31, 37, 43, 47, 53, 59, 67, 71, 73, 79, 83, 97, 103, 107, 113, 127, 131, 137, 139, 157, 163, 167, 173, 179, 193, 197, 223, 229, 233, 239, 263, 269, 271, 283, 293, 307, 311, 313, 317, 331, 337, 347, 353, 359, 367, 373, 379, 383, 397, 431, 433, 439, 443, 457, 463, 467, 479, 487, 503, 509, 523, 547, 563, 569, 571, 593, 599, 607, 613, 617, 631, 643, 647, 653, 659, 673, 677, 683, 701, 709, 719, 733, 739, 743, 751, 761, 769, 773, 797, 823, 829, 839, 853, 859, 863, 883, 907, 929, 937, 947, 953, 967, 971, 977, 983, 997

11

211 = 2, 311 = 3, 511 = 5, 711 = 7, 1211 = 13, 1611 = 17, 2111 = 23, 2711 = 29, 2911 = 31, 3411 = 37, 3A11 = 43, 4311 = 47, 4911 = 53, 5411 = 59, 6111 = 67, 6511 = 71, 6711 = 73, 7211 = 79, 7611 = 83, 8911 = 97, 9411 = 103, 9811 = 107, A311 = 113, 10611 = 127, 10A11 = 131, 11511 = 137, 11711 = 139, 13311 = 157, 13911 = 163, 14211 = 167, 14811 = 173, 15311 = 179, 16611 = 193, 16A11 = 197, 19311 = 223, 19911 = 229, 1A211 = 233, 1A811 = 239, 21A11 = 263, 22511 = 269, 22711 = 271, 23811 = 283, 24711 = 293, 25A11 = 307, 26311 = 311, 26511 = 313, 26911 = 317, 28111 = 331, 28711 = 337, 29611 = 347, 2A111 = 353, 2A711 = 359, 30411 = 367, 30A11 = 373, 31511 = 379, 31911 = 383, 33111 = 397, 36211 = 431, 36411 = 433, 36A11 = 439, 37311 = 443, 38611 = 457, 39111 = 463, 39511 = 467, 3A611 = 479, 40311 = 487, 41811 = 503, 42311 = 509, 43611 = 523, 45811 = 547, 47211 = 563, 47811 = 569, 47A11 = 571, 49A11 = 593, 4A511 = 599, 50211 = 607, 50811 = 613, 51111 = 617, 52411 = 631, 53511 = 643, 53911 = 647, 54411 = 653, 54A11 = 659, 56211 = 673, 56611 = 677, 57111 = 683, 58811 = 701, 59511 = 709, 5A411 = 719, 60711 = 733, 61211 = 739, 61611 = 743, 62311 = 751, 63211 = 761, 63A11 = 769, 64311 = 773, 66511 = 797, 68911 = 823, 69411 = 829, 6A311 = 839, 70611 = 853, 71111 = 859, 71511 = 863, 73311 = 883, 75511 = 907, 77511 = 929, 78211 = 937, 79111 = 947, 79711 = 953, 7AA11 = 967, 80311 = 971, 80911 = 977, 81411 = 983, 82711 = 997

12

212 = 2, 312 = 3, 512 = 5, 712 = 7, B12 = 11, 1512 = 17, 1712 = 19, 1B12 = 23, 2512 = 29, 2712 = 31, 3112 = 37, 3512 = 41, 3712 = 43, 3B12 = 47, 4512 = 53, 4B12 = 59, 5112 = 61, 5712 = 67, 5B12 = 71, 6712 = 79, 6B12 = 83, 7512 = 89, 8512 = 101, 8712 = 103, 8B12 = 107, 9512 = 113, A712 = 127, AB12 = 131, B512 = 137, B712 = 139, 10512 = 149, 10712 = 151, 11712 = 163, 11B12 = 167, 12512 = 173, 12B12 = 179, 13112 = 181, 13B12 = 191, 14512 = 197, 14712 = 199, 15712 = 211, 16712 = 223, 16B12 = 227, 17112 = 229, 17512 = 233, 17B12 = 239, 18B12 = 251, 19512 = 257, 19B12 = 263, 1A512 = 269, 1A712 = 271, 1B112 = 277, 1B512 = 281, 1B712 = 283, 20512 = 293, 21712 = 307, 21B12 = 311, 22512 = 317, 23712 = 331, 24B12 = 347, 25112 = 349, 25512 = 353, 25B12 = 359, 26712 = 367, 27112 = 373, 27712 = 379, 27B12 = 383, 28512 = 389, 29512 = 401, 2AB12 = 419, 30112 = 433, 30712 = 439, 30B12 = 443, 31512 = 449, 32112 = 457, 32512 = 461, 32712 = 463, 32B12 = 467, 33B12 = 479, 34712 = 487, 34B12 = 491, 35712 = 499, 35B12 = 503, 36512 = 509, 37512 = 521, 37712 = 523, 39112 = 541, 39712 = 547, 3A512 = 557, 3AB12 = 563, 3B512 = 569, 3B712 = 571, 40B12 = 587, 41512 = 593, 41B12 = 599, 42712 = 607, 43112 = 613, 43512 = 617, 43712 = 619, 45512 = 641, 45712 = 643, 45B12 = 647, 46512 = 653, 46B12 = 659, 48512 = 677, 48B12 = 683, 4A512 = 701, 4B112 = 709, 4BB12 = 719, 50712 = 727, 51112 = 733, 51712 = 739, 51B12 = 743, 52712 = 751, 53112 = 757, 53512 = 761, 54112 = 769, 54512 = 773, 55712 = 787, 57512 = 809, 57712 = 811, 58512 = 821, 58712 = 823, 58B12 = 827, 59112 = 829, 59B12 = 839, 5B112 = 853, 5B512 = 857, 5B712 = 859, 5BB12 = 863, 61512 = 881, 61712 = 883, 61B12 = 887, 63712 = 907, 63B12 = 911, 64712 = 919, 66B12 = 947, 67512 = 953, 68712 = 967, 68B12 = 971, 69512 = 977, 69B12 = 983, 6A712 = 991, 6B112 = 997

13

213 = 2, 313 = 3, 513 = 5, 713 = 7, B13 = 11, 2313 = 29, 2513 = 31, 2B13 = 37, 3213 = 41, 3413 = 43, 3813 = 47, 4713 = 59, 5213 = 67, 5613 = 71, 5813 = 73, 6513 = 83, 6B13 = 89, 7613 = 97, 7A13 = 101, 7C13 = 103, 8313 = 107, 8513 = 109, A713 = 137, B613 = 149, B813 = 151, C713 = 163, CB13 = 167, 20B13 = 349, 22313 = 367, 23213 = 379, 23613 = 383, 23C13 = 389, 24713 = 397, 24B13 = 401, 25613 = 409, 26313 = 419, 26513 = 421, 27A13 = 439, 2AB13 = 479, 2B613 = 487, 2BA13 = 491, 2C513 = 499, 30213 = 509, 32813 = 541, 34413 = 563, 34A13 = 569, 34C13 = 571, 36213 = 587, 36813 = 593, 38213 = 613, 38613 = 617, 38813 = 619, 3A413 = 641, 40713 = 683, 42713 = 709, 43413 = 719, 45213 = 743, 46713 = 761, 47213 = 769, 47613 = 773, 48713 = 787, 4B813 = 827, 4C713 = 839, 50813 = 853, 52613 = 877, 52A13 = 881, 52C13 = 883, 56613 = 929, 58413 = 953, 5A213 = 977, 5A813 = 983

14

214 = 2, 314 = 3, 514 = 5, 714 = 7, B14 = 11, D14 = 13, 1314 = 17, 1514 = 19, 2114 = 29, 2314 = 31, 2914 = 37, 2D14 = 41, 3114 = 43, 3514 = 47, 3B14 = 53, 4314 = 59, 4514 = 61, 4B14 = 67, 5114 = 71, 5314 = 73, 5914 = 79, 5D14 = 83, 6514 = 89, 6D14 = 97, 7314 = 101, 7514 = 103, 7914 = 107, 7B14 = 109, 9514 = 131, 9B14 = 137, 9D14 = 139, AB14 = 151, B314 = 157, B914 = 163, BD14 = 167, C514 = 173, CB14 = 179, CD14 = 181, D914 = 191, DB14 = 193, 10314 = 199, 12314 = 227, 12514 = 229, 12914 = 233, 13114 = 239, 13314 = 241, 13D14 = 251, 14514 = 257, 14B14 = 263, 15314 = 269, 15514 = 271, 15B14 = 277, 16314 = 283, 16D14 = 293, 18314 = 311, 18514 = 313, 1AB14 = 347, 1B314 = 353, 1B914 = 359, 1C314 = 367, 1D514 = 383, 1DB14 = 389, 20514 = 397, 20914 = 401, 21314 = 409, 21D14 = 419, 22114 = 421, 22D14 = 433, 23514 = 439, 23914 = 443, 24114 = 449, 24914 = 457, 24D14 = 461, 25114 = 463, 26314 = 479, 27114 = 491, 27914 = 499, 27D14 = 503, 29314 = 521, 29514 = 523, 2A914 = 541, 2B114 = 547, 2C314 = 563, 2C914 = 569, 2CB14 = 571, 2D314 = 577, 2DD14 = 587, 30514 = 593, 30B14 = 599, 30D14 = 601, 31514 = 607, 31B14 = 613, 32114 = 617, 32314 = 619, 33114 = 631, 33B14 = 641, 34314 = 647, 35114 = 659, 35314 = 661, 36114 = 673, 36514 = 677, 36B14 = 683, 37514 = 691, 38114 = 701, 39514 = 719, 39D14 = 727, 3A514 = 733, 3AB14 = 739, 3B114 = 743, 3B914 = 751, 3C114 = 757, 3C514 = 761, 3CD14 = 769, 40314 = 787, 40D14 = 797, 41B14 = 809, 42914 = 821, 42B14 = 823, 43114 = 827, 43314 = 829, 43D14 = 839, 45314 = 857, 45514 = 859, 45914 = 863, 46D14 = 881, 47514 = 887, 48B14 = 907, 4A514 = 929, 4B314 = 941, 4B914 = 947, 4D514 = 971, 4DB14 = 977, 50314 = 983, 50B14 = 991, 51314 = 997

15

215 = 2, 315 = 3, 515 = 5, 715 = 7, B15 = 11, D15 = 13, 1215 = 17, 2115 = 31, 2715 = 37, 2B15 = 41, 2D15 = 43, 3215 = 47, 3815 = 53, 3E15 = 59, 4715 = 67, 4B15 = 71, 4D15 = 73, 5415 = 79, 5815 = 83, 5E15 = 89, 6715 = 97, 6B15 = 101, 6D15 = 103, 7215 = 107, 7415 = 109, 7815 = 113, 8715 = 127, 8B15 = 131, 9215 = 137, A715 = 157, AD15 = 163, B215 = 167, B815 = 173, BE15 = 179, CB15 = 191, CD15 = 193, D215 = 197, D415 = 199, ED15 = 223, 10215 = 227, 12215 = 257, 12815 = 263, 12E15 = 269, 16215 = 317, 18215 = 347, 20715 = 457, 20B15 = 461, 20D15 = 463, 21215 = 467, 21E15 = 479, 22715 = 487, 22B15 = 491, 23815 = 503, 23E15 = 509, 24B15 = 521, 24D15 = 523, 26115 = 541, 26715 = 547, 27215 = 557, 27815 = 563, 27E15 = 569, 28115 = 571, 28715 = 577, 29215 = 587, 2A115 = 601, 2A715 = 607, 2AD15 = 613, 2B215 = 617, 2B415 = 619, 2C115 = 631, 2CB15 = 641, 2CD15 = 643, 2D215 = 647, 2D815 = 653, 2DE15 = 659, 2E115 = 661, 2ED15 = 673, 30215 = 677, 30815 = 683, 32415 = 709, 32E15 = 719, 34815 = 743, 36815 = 773, 38215 = 797, 38E15 = 809, 3A215 = 827, 3AE15 = 839, 3C215 = 857, 3C815 = 863, 3E215 = 887, 40715 = 907, 40B15 = 911, 42715 = 937, 42B15 = 941, 43215 = 947, 43815 = 953, 44715 = 967, 44B15 = 971, 45815 = 983, 46715 = 997

16

216 = 2, 316 = 3, 516 = 5, 716 = 7, B16 = 11, D16 = 13, 1316 = 19, 1716 = 23, 1D16 = 29, 2516 = 37, 2916 = 41, 2B16 = 43, 2F16 = 47, 3516 = 53, 3B16 = 59, 3D16 = 61, 4316 = 67, 4716 = 71, 5316 = 83, 5916 = 89, 6516 = 101, 6716 = 103, 6B16 = 107, 6D16 = 109, 7116 = 113, 7F16 = 127, 8316 = 131, 8B16 = 139, 9516 = 149, 9716 = 151, 9D16 = 157, A316 = 163, A716 = 167, AD16 = 173, B316 = 179, B516 = 181, BF16 = 191, C516 = 197, C716 = 199, D316 = 211, DF16 = 223, E316 = 227, E516 = 229, FB16 = 251, 10716 = 263, 10D16 = 269, 12516 = 293, 13316 = 307, 13716 = 311, 13916 = 313, 13D16 = 317, 15D16 = 349, 16716 = 359, 17516 = 373, 17B16 = 379, 17F16 = 383, 18D16 = 397, 1A316 = 419, 1B716 = 439, 1CD16 = 461, 1D316 = 467, 1DF16 = 479, 1E716 = 487, 1F316 = 499, 1F716 = 503, 1FD16 = 509, 20916 = 521, 20B16 = 523, 21D16 = 541, 23916 = 569, 23B16 = 571, 24B16 = 587, 25116 = 593, 25716 = 599, 25916 = 601, 25F16 = 607, 26516 = 613, 26916 = 617, 26B16 = 619, 28316 = 643, 29316 = 659, 29516 = 661, 2A516 = 677, 2AB16 = 683, 2B316 = 691, 2BD16 = 701, 2C516 = 709, 2CF16 = 719, 2E316 = 739, 2EF16 = 751, 2F516 = 757, 2F916 = 761, 30516 = 773, 31316 = 787, 31D16 = 797, 32916 = 809, 32B16 = 811, 33516 = 821, 33B16 = 827, 33D16 = 829, 34716 = 839, 35516 = 853, 35916 = 857, 35B16 = 859, 35F16 = 863, 36D16 = 877, 37116 = 881, 38B16 = 907, 39716 = 919, 3AD16 = 941, 3B316 = 947, 3B916 = 953, 3C716 = 967, 3CB16 = 971, 3D116 = 977, 3D716 = 983, 3DF16 = 991, 3E516 = 997

17

217 = 2, 317 = 3, 517 = 5, 717 = 7, B17 = 11, D17 = 13, 1217 = 19, 2317 = 37, 2717 = 41, 2917 = 43, 2D17 = 47, 3217 = 53, 3817 = 59, 3A17 = 61, 3G17 = 67, 4317 = 71, 4517 = 73, 4B17 = 79, 5417 = 89, 5C17 = 97, 5G17 = 101, 6517 = 107, 6717 = 109, 6B17 = 113, 7817 = 127, 7C17 = 131, 8317 = 139, 8D17 = 149, A317 = 173, AB17 = 181, B417 = 191, B617 = 193, BA17 = 197, BC17 = 199, C717 = 211, D217 = 223, D617 = 227, D817 = 229, DC17 = 233, E317 = 241, ED17 = 251, F217 = 257, G517 = 277, GB17 = 283, 12817 = 331, 12E17 = 337, 14217 = 359, 1A217 = 461, 1G217 = 563, 20917 = 587, 22717 = 619, 23217 = 631, 23C17 = 641, 23E17 = 643, 24717 = 653, 24D17 = 659, 26317 = 683, 26B17 = 691, 27417 = 701, 27C17 = 709, 28D17 = 727, 29217 = 733, 29817 = 739, 29C17 = 743, 2A317 = 751, 2A917 = 757, 2AD17 = 761, 2B417 = 769, 2DA17 = 809, 2DC17 = 811, 2E717 = 823, 2ED17 = 829, 2G317 = 853, 2G717 = 857, 2G917 = 859, 2GD17 = 863, 30A17 = 877, 30G17 = 883, 32617 = 907, 32A17 = 911, 34217 = 937, 36217 = 971, 36817 = 977

18

218 = 2, 318 = 3, 518 = 5, 718 = 7, B18 = 11, D18 = 13, H18 = 17, 1518 = 23, 1B18 = 29, 1D18 = 31, 2118 = 37, 2518 = 41, 2718 = 43, 2B18 = 47, 2H18 = 53, 3518 = 59, 3718 = 61, 3D18 = 67, 3H18 = 71, 4718 = 79, 4B18 = 83, 4H18 = 89, 5718 = 97, 5B18 = 101, 5D18 = 103, 5H18 = 107, 6518 = 113, 7118 = 127, 7518 = 131, 7B18 = 137, 7D18 = 139, 8518 = 149, 8718 = 151, 8D18 = 157, 9518 = 167, 9B18 = 173, 9H18 = 179, AB18 = 191, AD18 = 193, AH18 = 197, B118 = 199, BD18 = 211, C718 = 223, CB18 = 227, CD18 = 229, CH18 = 233, D518 = 239, D718 = 241, DH18 = 251, E518 = 257, EB18 = 263, EH18 = 269, F718 = 277, FB18 = 281, FD18 = 283, G518 = 293, H118 = 307, H518 = 311, H718 = 313, HB18 = 317, 10718 = 331, 10D18 = 337, 11518 = 347, 11B18 = 353, 12718 = 367, 12D18 = 373, 13518 = 383, 13B18 = 389, 14518 = 401, 14D18 = 409, 15518 = 419, 15718 = 421, 15H18 = 431, 16B18 = 443, 17B18 = 461, 17D18 = 463, 18B18 = 479, 19518 = 491, 19D18 = 499, 19H18 = 503, 1A518 = 509, 1AH18 = 521, 1B118 = 523, 1C718 = 547, 1CH18 = 557, 1D518 = 563, 1DB18 = 569, 1DD18 = 571, 1EB18 = 587, 1EH18 = 593, 1F518 = 599, 1F718 = 601, 1FD18 = 607, 1G518 = 617, 1H118 = 631, 1HB18 = 641, 1HD18 = 643, 20518 = 653, 20B18 = 659, 20D18 = 661, 21718 = 673, 21B18 = 677, 21H18 = 683, 22718 = 691, 22H18 = 701, 23718 = 709, 23H18 = 719, 24718 = 727, 25118 = 739, 25518 = 743, 25D18 = 751, 26118 = 757, 26518 = 761, 26H18 = 773, 27D18 = 787, 28518 = 797, 28H18 = 809, 29118 = 811, 29B18 = 821, 29H18 = 827, 2A118 = 829, 2AB18 = 839, 2B718 = 853, 2BB18 = 857, 2BD18 = 859, 2BH18 = 863, 2CD18 = 877, 2CH18 = 881, 2D118 = 883, 2D518 = 887, 2E718 = 907, 2EB18 = 911, 2F118 = 919, 2FB18 = 929, 2G118 = 937, 2G518 = 941, 2GB18 = 947, 2GH18 = 953, 2HD18 = 967, 2HH18 = 971, 30518 = 977, 30B18 = 983, 31718 = 997

19

219 = 2, 319 = 3, 519 = 5, 719 = 7, B19 = 11, D19 = 13, H19 = 17, 2319 = 41, 2519 = 43, 2919 = 47, 2F19 = 53, 3219 = 59, 3419 = 61, 3A19 = 67, 3E19 = 71, 3G19 = 73, 4319 = 79, 4719 = 83, 4D19 = 89, 5219 = 97, 5619 = 101, 5819 = 103, 5C19 = 107, 5E19 = 109, 5I19 = 113, 6D19 = 127, 6H19 = 131, 7419 = 137, 7619 = 139, 7G19 = 149, 7I19 = 151, 8519 = 157, 8B19 = 163, 9219 = 173, A319 = 193, A719 = 197, B219 = 211, BE19 = 223, BI19 = 227, C519 = 233, CB19 = 239, CD19 = 241, D419 = 251, DA19 = 257, DG19 = 263, E319 = 269, E519 = 271, EB19 = 277, EH19 = 283, G319 = 307, G719 = 311, GD19 = 317, H819 = 331, HE19 = 337, I519 = 347, I719 = 349, IB19 = 353, IH19 = 359, 20519 = 727, 20B19 = 733, 20H19 = 739, 22919 = 769, 23819 = 787, 23I19 = 797, 24D19 = 811, 25419 = 821, 25619 = 823, 25A19 = 827, 25C19 = 829, 26319 = 839, 26H19 = 853, 27419 = 859, 28319 = 877, 28919 = 883, 29E19 = 907, 29I19 = 911, 2A719 = 919, 2C319 = 953, 2E319 = 991, 2E919 = 997

20

220 = 2, 320 = 3, 520 = 5, 720 = 7, B20 = 11, D20 = 13, H20 = 17, J20 = 19, 1320 = 23, 1B20 = 31, 1H20 = 37, 2120 = 41, 2320 = 43, 2720 = 47, 2D20 = 53, 2J20 = 59, 3120 = 61, 3720 = 67, 3B20 = 71, 3D20 = 73, 3J20 = 79, 4320 = 83, 4H20 = 97, 5120 = 101, 5320 = 103, 5720 = 107, 5920 = 109, 5D20 = 113, 6720 = 127, 6B20 = 131, 6H20 = 137, 6J20 = 139, 7920 = 149, 7B20 = 151, 7H20 = 157, 8320 = 163, 8720 = 167, 8D20 = 173, 8J20 = 179, 9B20 = 191, 9D20 = 193, 9H20 = 197, 9J20 = 199, AB20 = 211, B320 = 223, B720 = 227, B920 = 229, BD20 = 233, BJ20 = 239, CB20 = 251, CH20 = 257, D320 = 263, D920 = 269, DB20 = 271, DH20 = 277, E320 = 283, ED20 = 293, F720 = 307, FB20 = 311, FD20 = 313, FH20 = 317, GB20 = 331, GH20 = 337, H720 = 347, H920 = 349, HD20 = 353, HJ20 = 359, I720 = 367, ID20 = 373, IJ20 = 379, J320 = 383, J920 = 389, JH20 = 397, 10J20 = 419, 11B20 = 431, 12320 = 443, 12H20 = 457, 13120 = 461, 13320 = 463, 13720 = 467, 13J20 = 479, 14B20 = 491, 15320 = 503, 15920 = 509, 16320 = 523, 17H20 = 557, 18320 = 563, 18B20 = 571, 18H20 = 577, 19D20 = 593, 19J20 = 599, 1AH20 = 617, 1BB20 = 631, 1C320 = 643, 1DH20 = 677, 1E320 = 683, 1EB20 = 691, 1H320 = 743, 1HB20 = 751, 1HH20 = 757, 1ID20 = 773, 1JH20 = 797, 20B20 = 811, 21120 = 821, 21320 = 823, 21720 = 827, 21920 = 829, 21J20 = 839, 22D20 = 853, 22J20 = 859, 23320 = 863, 23H20 = 877, 24120 = 881, 24320 = 883, 24720 = 887, 25720 = 907, 25J20 = 919, 26H20 = 937, 27120 = 941, 27720 = 947, 27D20 = 953, 28720 = 967, 29320 = 983, 29B20 = 991, 29H20 = 997

 

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