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Primi invertibili

Rappresentazione dei numeri 

Si chimano “invertibili” i numeri primi che rimangono tali se letti invertendo l’ordine delle cifre, come 107, che diventa 701. Alcuni Autori preferiscono escludere i primi palindromi.

Chiaramente non possono iniziare né con una cifra pari né con 5.

 

I primi invertibili minori di 1000 (escludendo quelli palindromi) sono: 13, 17, 31, 37, 71, 73, 79, 97, 107, 113, 149, 157, 167, 179, 199, 311, 337, 347, 359, 389, 701, 709, 733, 739, 743, 751, 761, 769, 907, 937, 941, 953, 967, 971, 983, 991.

Qui trovate i primi invertibili non palindromi fino a 107 (M. Fiorentini, 2017).

 

La tabella seguente riporta le minime sequenze di n primi invertibili consecutivi, per n fino a 12.

n

Minima sequenza

Autore

1

2

 

2

2, 3

 

3

2, 3, 5

 

4

2, 3, 5, 7

 

5

2, 3, 5, 7, 11

 

6

2, 3, 5, 7, 11, 13

 

7

2, 3, 5, 7, 11, 13, 17

 

8

727, 733, 739, 743, 751, 757, 761, 769

 

9

1193, 1201, 1213, 1217, 1223, 1229, 1231, 1237, 1249

Carlos Rivera, 1998

10

1193, 1201, 1213, 1217, 1223, 1229, 1231, 1237, 1249, 1259

Carlos Rivera, 1998

11

1477271183, 1477271249, 1477271251, 1477271269, 1477271291, 1477271311, 1477271317, 1477271351, 1477271357, 1477271381, 1477271387

Jud McCranie, 1999

12

9387802769, 9387802807, 9387802817, 9387802861, 9387802867, 9387802873, 9387802909, 9387802937, 9387802939, 9387802973, 9387802987, 9387803003

Jud McCranie, 1999

13

15423094826093, 15423094826147, 15423094826149, 15423094826197, 15423094826213, 15423094826257, 15423094826267, 15423094826287, 15423094826339, 15423094826341, 15423094826351, 15423094826369, 15423094826389

Giovanni Resta, 2011

 

Le massime coppie note sono:

  • 101014 + 2888601 • 10504 + 1 e 101014 +1068882 • 10504 + 1 (H. Dubner, 1015 cifre);

  • 102003 + (n – 999999999) • 10997 – 1 e 102003 + (R(n) – 999999999) • 10997 – 1, dove R(n) indica n scritto al contrario, per n uguale a 107671501, 158957701, 168586801, 268720301, 292300601, 318811301, 689715601, 856978001, 996358669 (Jens Kruse Andersen, 2007, 2003 cifre);

  • 103929 – 571624798 • 101960 – 1 e 103929 – 897426175 • 101960 – 1 (Jens Kruse Andersen, 2003, 3929 cifre; da notare che 3929 è a sua volta un primo invertibile);

  • 105013 + 103296 + 101834 + 1 e 105013 + 103179 + 101717 + 1 (Jens Kruse Andersen, 2011, 5014 cifre);

  • 1010006 + 941992101 • 104999 + 1 e 1010006 + 101299149 • 104999 + 1 (Jens Kruse Andersen, 2007, 10007 cifre).

 

La massima coppia nota di primi pandigitali invertibili si deve a H. Dubner ed è costituita da 101006 + 234567890629590 • 10496 + 1 e 101006 + 95926098765432 • 10496 + 1.

 

Si può dimostrare che l’unico caso in cui i primi invertibili p e q di una coppia possono essere legati da una relazione del tipo q = kp ± 1 è quello in cui q = 2p – 1; in tal caso inoltre p deve essere della forma 3(9)n7. L’unica coppia nota di questo genere è quella formata da 37 e 73, se ve ne sono altre, hanno più di 10000 cifre (M. Fiorentini, 2017).

 

Il minimo quadrato magico formato da primi invertibili è il seguente.

17207

18731

16493

16763

17477

18191

18461

16223

17747

 

La definizione può essere estesa ad altre basi; la tabella seguente riporta i primi invertibili fino a 1000 nelle basi fino a 20.

Base

Primi invertibili

2

10112 = 11, 11012 = 13, 101112 = 23, 111012 = 29, 1001012 = 37, 1010012 = 41, 1010112 = 43, 1011112 = 47, 1101012 = 53, 1111012 = 61, 10000112 = 67, 10001112 = 71, 10100112 = 83, 11000012 = 97, 11001012 = 101, 11100012 = 113, 100000112 = 131, 100101112 = 151, 101000112 = 163, 101001112 = 167, 101011012 = 173, 101101012 = 181, 110000012 = 193, 110001012 = 197, 110001112 = 199, 110111112 = 223, 111000112 = 227, 111001012 = 229, 111010012 = 233, 111110112 = 251, 1000001112 = 263, 1000011012 = 269, 1000101012 = 277, 1000110112 = 283, 1001100112 = 307, 1010010112 = 331, 1010100012 = 337, 1010111012 = 349, 1011000012 = 353, 1011001112 = 359, 1011101012 = 373, 1011111112 = 383, 1100110012 = 409, 1101001012 = 421, 1101011112 = 431, 1101100012 = 433, 1110000012 = 449, 1110011012 = 461, 1110011112 = 463, 1110111112 = 479, 1111001112 = 487, 1111010112 = 491, 1111101112 = 503, 1111111012 = 509, 10000010012 = 521, 10001110112 = 571, 10010000012 = 577, 10010101112 = 599, 10010110012 = 601, 10011010012 = 617, 10011010112 = 619, 10011101112 = 631, 10100000112 = 643, 10100011012 = 653, 10100101012 = 661, 10101001012 = 677, 10101010112 = 683, 10101100112 = 691, 10101111012 = 701, 10110001012 = 709, 10110101112 = 727, 10111000112 = 739, 10111101012 = 757, 11000001012 = 773, 11000111012 = 797, 11001101012 = 821, 11001101112 = 823, 11001110112 = 827, 11010001112 = 839, 11010101012 = 853, 11010110012 = 857, 11011100012 = 881, 11011100112 = 883, 11100010112 = 907, 11100011112 = 911, 11101010012 = 937, 11101011012 = 941, 11101100112 = 947, 11101110012 = 953, 11110001112 = 967

3

123 = 5, 213 = 7, 1023 = 11, 2013 = 19, 10113 = 31, 11013 = 37, 11123 = 41, 12023 = 47, 12223 = 53, 20213 = 61, 21113 = 67, 22213 = 79, 100023 = 83, 102023 = 101, 102113 = 103, 110123 = 113, 112013 = 127, 112123 = 131, 200013 = 163, 200123 = 167, 201223 = 179, 202013 = 181, 210023 = 191, 210113 = 193, 212113 = 211, 221023 = 227, 1002023 = 263, 1011023 = 281, 1012123 = 293, 1021123 = 311, 1100213 = 331, 1102213 = 349, 1110023 = 353, 1110223 = 359, 1122123 = 401, 1200113 = 409, 1201123 = 419, 1201213 = 421, 1202223 = 431, 1210213 = 439, 1211223 = 449, 1220113 = 463, 1220223 = 467, 2000123 = 491, 2001113 = 499, 2001223 = 503, 2010223 = 521, 2011013 = 523, 2020013 = 541, 2022123 = 563, 2100023 = 569, 2110213 = 601, 2112013 = 613, 2121013 = 631, 2122023 = 641, 2122113 = 643, 2201023 = 659, 2201113 = 661, 2202213 = 673, 2210023 = 677, 2211213 = 691, 2212223 = 701, 2220213 = 709, 2221223 = 719, 10001013 = 739, 10001123 = 743, 10011223 = 773, 10022223 = 809, 10100013 = 811, 10101023 = 821, 10101113 = 823, 10112023 = 857, 10112113 = 859, 10112223 = 863, 10121113 = 877, 10121223 = 881, 10220023 = 947, 10220223 = 953, 10222223 = 971, 11001023 = 983, 11002213 = 997

4

134 = 7, 314 = 13, 1134 = 23, 1334 = 31, 3114 = 53, 3314 = 61, 10034 = 67, 10214 = 73, 10334 = 79, 11034 = 83, 11214 = 89, 12014 = 97, 12114 = 101, 12234 = 107, 30014 = 193, 30114 = 197, 30134 = 199, 31034 = 211, 32214 = 233, 32334 = 239, 33014 = 241, 33234 = 251, 100314 = 269, 101114 = 277, 101214 = 281, 102114 = 293, 103134 = 311, 103214 = 313, 111014 = 337, 112014 = 353, 112334 = 367, 113334 = 383, 120314 = 397, 121014 = 401, 122034 = 419, 123014 = 433, 123234 = 443, 130014 = 449, 130214 = 457, 131034 = 467, 132134 = 487, 132234 = 491, 133034 = 499, 301314 = 797, 302214 = 809, 303134 = 823, 303234 = 827, 303314 = 829, 311234 = 859, 311334 = 863, 312314 = 877, 313014 = 881, 313034 = 883, 320334 = 911, 321134 = 919, 322314 = 941, 323034 = 947, 323214 = 953, 330234 = 971, 331134 = 983, 332114 = 997

5

125 = 7, 215 = 11, 235 = 13, 325 = 17, 345 = 19, 435 = 23, 1045 = 29, 1225 = 37, 1425 = 47, 2145 = 59, 2215 = 61, 2415 = 71, 2435 = 73, 3045 = 79, 3245 = 89, 3425 = 97, 4015 = 101, 4035 = 103, 4125 = 107, 4235 = 113, 10025 = 127, 10115 = 131, 10445 = 149, 11015 = 151, 11125 = 157, 11235 = 163, 11325 = 167, 12315 = 191, 12335 = 193, 13215 = 211, 13435 = 223, 14025 = 227, 14045 = 229, 14135 = 233, 14245 = 239, 20015 = 251, 20125 = 257, 20345 = 269, 20415 = 271, 21025 = 277, 21115 = 281, 21335 = 293, 22325 = 317, 23115 = 331, 23225 = 337, 23425 = 347, 23445 = 349, 24035 = 353, 24145 = 359, 24325 = 367, 30045 = 379, 30425 = 397, 31415 = 421, 32115 = 431, 32245 = 439, 32335 = 443, 32445 = 449, 33125 = 457, 33215 = 461, 33235 = 463, 34045 = 479, 34315 = 491, 40035 = 503, 40415 = 521, 40435 = 523, 41425 = 547, 42235 = 563, 42415 = 571, 43025 = 577, 43445 = 599, 44015 = 601, 44235 = 613, 44325 = 617, 44345 = 619, 100115 = 631, 100335 = 643, 101035 = 653, 102025 = 677, 102315 = 691, 104245 = 739, 104335 = 743, 110015 = 751, 110125 = 757, 110215 = 761, 110435 = 773, 111225 = 787, 112415 = 821, 114125 = 857, 114145 = 859, 120115 = 881, 120135 = 883, 120225 = 887, 122315 = 941, 122425 = 947, 124025 = 977, 124135 = 983, 124425 = 997

6

156 = 11, 516 = 31, 1056 = 41, 1256 = 53, 1356 = 59, 1556 = 71, 5016 = 181, 5216 = 193, 5316 = 199, 5516 = 211, 10156 = 227, 10256 = 233, 11056 = 257, 11156 = 263, 11316 = 271, 11416 = 277, 11456 = 281, 12056 = 293, 12316 = 307, 12356 = 311, 12416 = 313, 12456 = 317, 13116 = 331, 13216 = 337, 13416 = 349, 13556 = 359, 14116 = 367, 14216 = 373, 14316 = 379, 14356 = 383, 14456 = 389, 15556 = 431

7

147 = 11, 167 = 13, 237 = 17, 257 = 19, 327 = 23, 417 = 29, 527 = 37, 567 = 41, 617 = 43, 657 = 47, 1047 = 53, 1247 = 67, 1427 = 79, 1557 = 89, 1667 = 97, 2037 = 101, 2417 = 127, 2547 = 137, 2567 = 139, 3027 = 149, 3047 = 151, 3267 = 167, 3357 = 173, 3447 = 179, 3647 = 193, 4017 = 197, 4037 = 199, 4217 = 211, 4437 = 227, 4457 = 229, 4527 = 233, 4637 = 241, 5337 = 269, 5447 = 277, 5517 = 281, 6237 = 311, 6527 = 331, 6617 = 337, 10047 = 347, 10427 = 373, 10647 = 389, 11057 = 397, 11127 = 401, 11367 = 419, 11417 = 421, 11567 = 433, 11657 = 439, 12027 = 443, 12357 = 467, 12537 = 479, 13127 = 499, 13167 = 503, 13457 = 523, 14117 = 547, 14247 = 557, 14427 = 569, 14447 = 571, 14537 = 577, 14667 = 587, 15167 = 601, 15617 = 631, 16227 = 653, 16517 = 673, 16557 = 677, 16647 = 683, 20217 = 701, 20457 = 719, 20657 = 733, 21047 = 739, 21117 = 743, 21317 = 757, 21357 = 761, 21537 = 773, 22367 = 811, 22527 = 821, 22617 = 827, 23067 = 839, 23267 = 853, 24017 = 883, 24347 = 907, 24417 = 911, 24657 = 929, 25227 = 947, 25557 = 971, 26237 = 997

8

158 = 13, 358 = 29, 378 = 31, 518 = 41, 538 = 43, 578 = 47, 738 = 59, 758 = 61, 1038 = 67, 1078 = 71, 1178 = 79, 1458 = 101, 1538 = 107, 3018 = 193, 3518 = 233, 3578 = 239, 5418 = 353, 5778 = 383, 7018 = 449, 7118 = 457, 7538 = 491, 7758 = 509, 10118 = 521, 10138 = 523, 10358 = 541, 11018 = 577, 11138 = 587, 12038 = 643, 12458 = 677, 12538 = 683, 13178 = 719, 13358 = 733, 13578 = 751, 13658 = 757, 14058 = 773, 14238 = 787, 14538 = 811, 14678 = 823, 14738 = 827, 14758 = 829, 15258 = 853, 15618 = 881, 16178 = 911, 16278 = 919, 16518 = 937, 16718 = 953, 17138 = 971, 17278 = 983, 17378 = 991

9

129 = 11, 149 = 13, 189 = 17, 219 = 19, 259 = 23, 419 = 37, 479 = 43, 529 = 47, 749 = 67, 789 = 71, 819 = 73, 879 = 79, 1029 = 83, 1179 = 97, 1229 = 101, 1359 = 113, 1649 = 139, 1779 = 151, 1849 = 157, 2019 = 163, 2189 = 179, 2219 = 181, 2349 = 193, 2389 = 197, 2749 = 229, 2859 = 239, 2879 = 241, 4079 = 331, 4329 = 353, 4479 = 367, 4619 = 379, 4659 = 383, 4729 = 389, 4819 = 397, 5319 = 433, 5379 = 439, 5579 = 457, 5649 = 463, 5829 = 479, 7049 = 571, 7119 = 577, 7289 = 593, 7359 = 599, 7449 = 607, 7559 = 617, 7719 = 631, 7829 = 641, 7889 = 647, 8129 = 659, 8279 = 673, 8329 = 677, 8879 = 727, 10049 = 733, 10119 = 739, 10159 = 743, 10249 = 751, 10359 = 761, 10889 = 809, 11019 = 811, 11129 = 821, 11189 = 827, 11479 = 853, 11529 = 857, 11549 = 859, 11589 = 863, 12319 = 919, 12429 = 929, 12519 = 937, 12689 = 953, 12849 = 967, 13059 = 977, 13129 = 983, 13219 = 991

10

13, 17, 31, 37, 71, 73, 79, 97, 107, 113, 149, 157, 167, 179, 199, 311, 337, 347, 359, 389, 701, 709, 733, 739, 743, 751, 761, 769, 907, 937, 941, 953, 967, 971, 983, 991

11

1211 = 13, 1611 = 17, 1811 = 19, 2111 = 23, 2711 = 29, 2911 = 31, 3411 = 37, 3A11 = 43, 4311 = 47, 4911 = 53, 5611 = 61, 6111 = 67, 6511 = 71, 6711 = 73, 7211 = 79, 7611 = 83, 8111 = 89, 8911 = 97, 9211 = 101, 9411 = 103, 9811 = 107, A311 = 113, 10611 = 127, 11511 = 137, 11711 = 139, 12811 = 151, 13311 = 157, 13911 = 163, 14811 = 173, 15311 = 179, 15511 = 181, 16A11 = 197, 18211 = 211, 19311 = 223, 19711 = 227, 1A211 = 233, 1AA11 = 241, 20911 = 251, 21A11 = 263, 23611 = 281, 26311 = 311, 26511 = 313, 28111 = 331, 28711 = 337, 29611 = 347, 29811 = 349, 2A111 = 353, 30411 = 367, 30A11 = 373, 31511 = 379, 31911 = 383, 32411 = 389, 33111 = 397, 33511 = 401, 35111 = 419, 36211 = 431, 36A11 = 439, 38A11 = 461, 39111 = 463, 3A611 = 479, 40311 = 487, 41811 = 503, 42311 = 509, 47811 = 569, 47A11 = 571, 4A511 = 599, 51111 = 617, 51311 = 619, 53311 = 641, 54A11 = 659, 55111 = 661, 56211 = 673, 56611 = 677, 57911 = 691, 58811 = 701, 5A411 = 719, 60111 = 727, 60711 = 733, 62911 = 757, 63211 = 761, 63A11 = 769, 66511 = 797, 67811 = 811, 68711 = 821, 69211 = 827, 6A311 = 839, 70611 = 853, 70A11 = 857, 71111 = 859, 72811 = 877, 75911 = 911, 78211 = 937, 78611 = 941, 79111 = 947, 7AA11 = 967, 80911 = 977, 81411 = 983, 82111 = 991, 82711 = 997

12

1512 = 17, 5112 = 61, 5712 = 67, 5B12 = 71, 7512 = 89, B512 = 137, 10712 = 151, 11712 = 163, 11B12 = 167, 12B12 = 179, 13B12 = 191, 14512 = 197, 15712 = 211, 16B12 = 227, 17B12 = 239, 19512 = 257, 19B12 = 263, 1A712 = 271, 1B512 = 281, 50712 = 727, 51B12 = 743, 54112 = 769, 57712 = 811, 58712 = 823, 59112 = 829, 59B12 = 839, 5B112 = 853, 5BB12 = 863

13

1413 = 17, 1613 = 19, 1A13 = 23, 2313 = 29, 2513 = 31, 3213 = 41, 3813 = 47, 4113 = 53, 5213 = 67, 5613 = 71, 5813 = 73, 6113 = 79, 6513 = 83, 6B13 = 89, 7A13 = 101, 7C13 = 103, 8313 = 107, 8513 = 109, 9A13 = 127, A113 = 131, A713 = 137, A913 = 139, B613 = 149, C713 = 163, 10413 = 173, 10C13 = 181, 11B13 = 193, 13313 = 211, 14C13 = 233, 15513 = 239, 15713 = 241, 17313 = 263, 17913 = 269, 17B13 = 271, 19713 = 293, 1A813 = 307, 1C613 = 331, 20913 = 347, 20B13 = 349, 21813 = 359, 22913 = 373, 23C13 = 389, 24713 = 397, 24B13 = 401, 26313 = 419, 27413 = 433, 27A13 = 439, 28713 = 449, 29813 = 463, 32813 = 541, 33113 = 547, 33B13 = 557, 34C13 = 571, 36213 = 587, 36813 = 593, 37113 = 599, 38813 = 619, 39713 = 631, 3A413 = 641, 3AA13 = 647, 3B913 = 659, 40113 = 677, 40713 = 683, 42713 = 709, 45A13 = 751, 47213 = 769, 47613 = 773, 48713 = 787, 4A313 = 809, 4B813 = 827, 52A13 = 881, 54A13 = 907, 55113 = 911, 56613 = 929, 5A813 = 983, 5B913 = 997

14

1314 = 17, 1514 = 19, 1914 = 23, 3114 = 43, 3514 = 47, 3B14 = 53, 5114 = 71, 5314 = 73, 5914 = 79, 9114 = 127, 9514 = 131, 9B14 = 137, 9D14 = 139, B314 = 157, B914 = 163, BD14 = 167, D914 = 191, DB14 = 193, 12314 = 227, 12514 = 229, 13314 = 241, 13D14 = 251, 14B14 = 263, 15314 = 269, 15514 = 271, 16314 = 283, 16D14 = 293, 17D14 = 307, 18314 = 311, 18514 = 313, 18914 = 317, 1AB14 = 347, 1AD14 = 349, 1B314 = 353, 1C314 = 367, 1C914 = 373, 1D514 = 383, 1DB14 = 389, 30514 = 593, 30D14 = 601, 31514 = 607, 32114 = 617, 33114 = 631, 33D14 = 643, 34914 = 653, 35114 = 659, 36114 = 673, 36B14 = 683, 38114 = 701, 38914 = 709, 39514 = 719, 39D14 = 727, 3A514 = 733, 3B114 = 743, 3C114 = 757, 3C514 = 761, 3CD14 = 769, 50314 = 983, 50B14 = 991, 51314 = 997

15

1215 = 17, 1415 = 19, 1E15 = 29, 2115 = 31, 2715 = 37, 2B15 = 41, 2D15 = 43, 4115 = 61, 4715 = 67, 4D15 = 73, 7215 = 107, 7415 = 109, 7815 = 113, 8715 = 127, 8B15 = 131, B215 = 167, B815 = 173, D215 = 197, D415 = 199, E115 = 211, 10815 = 233, 12815 = 263, 12E15 = 269, 13715 = 277, 13B15 = 281, 13D15 = 283, 14815 = 293, 15B15 = 311, 15D15 = 313, 16215 = 317, 18215 = 347, 18415 = 349, 18E15 = 359, 19D15 = 373, 1A415 = 379, 1A815 = 383, 1AE15 = 389, 1B715 = 397, 1CE15 = 419, 1DB15 = 431, 1DD15 = 433, 1E815 = 443, 1EE15 = 449, 20B15 = 461, 20D15 = 463, 21E15 = 479, 22715 = 487, 23415 = 499, 23815 = 503, 26115 = 541, 26715 = 547, 27815 = 563, 27E15 = 569, 28115 = 571, 28715 = 577, 2CB15 = 641, 2D815 = 653, 2DE15 = 659, 2ED15 = 673, 40715 = 907, 41E15 = 929, 42715 = 937, 43215 = 947, 44B15 = 971, 45815 = 983, 46715 = 997

16

1716 = 23, 1F16 = 31, 3516 = 53, 3B16 = 59, 3D16 = 61, 5316 = 83, 5916 = 89, 7116 = 113, 9516 = 149, B316 = 179, BF16 = 191, D316 = 211, F116 = 241, FB16 = 251, 10D16 = 269, 11516 = 277, 11B16 = 283, 15B16 = 347, 16716 = 359, 18516 = 389, 18D16 = 397, 1A316 = 419, 1AF16 = 431, 1D316 = 467, 1DF16 = 479, 1E716 = 487, 1EB16 = 491, 1F316 = 499, 30516 = 773, 31D16 = 797, 32916 = 809, 32B16 = 811, 35F16 = 863, 37716 = 887, 3A116 = 929, 3A916 = 937, 3AD16 = 941, 3C716 = 967, 3CB16 = 971, 3D116 = 977, 3D716 = 983, 3DF16 = 991

17

1617 = 23, 1E17 = 31, 2317 = 37, 2D17 = 47, 3217 = 53, 3817 = 59, 3A17 = 61, 4517 = 73, 4B17 = 79, 5417 = 89, 5G17 = 101, 6117 = 103, 6B17 = 113, 7C17 = 131, 8317 = 139, 8D17 = 149, 8F17 = 151, 9A17 = 163, A317 = 173, A917 = 179, AB17 = 181, B417 = 191, B617 = 193, BA17 = 197, C717 = 211, D217 = 223, D817 = 229, E117 = 239, F817 = 263, G517 = 277, 12817 = 331, 14217 = 359, 15517 = 379, 15917 = 383, 15F17 = 389, 17B17 = 419, 17D17 = 421, 18617 = 431, 1A417 = 463, 1BF17 = 491, 1F317 = 547, 1FD17 = 557, 1GA17 = 571, 20F17 = 593, 21617 = 601, 22517 = 617, 23E17 = 643, 24117 = 647, 25E17 = 677, 26317 = 683, 27417 = 701, 28517 = 719, 29817 = 739, 29C17 = 743, 2A317 = 751, 2AD17 = 761, 2DC17 = 811, 2G717 = 857, 30E17 = 881, 32A17 = 911, 34C17 = 947, 35F17 = 967, 36217 = 971, 36817 = 977, 37517 = 991, 37B17 = 997

18

1B18 = 29, 5718 = 97, 5D18 = 103, 5H18 = 107, 7518 = 131, 7D18 = 139, B118 = 199, D518 = 239, D718 = 241, H518 = 311, 10718 = 331, 11718 = 349, 11B18 = 353, 11H18 = 359, 14518 = 401, 16718 = 439, 16B18 = 443, 17B18 = 461, 17D18 = 463, 18B18 = 479, 19518 = 491, 1A518 = 509, 1AH18 = 521, 1DD18 = 571, 1F718 = 601, 1FD18 = 607, 1G718 = 619, 1HD18 = 643

19

1A19 = 29, 1C19 = 31, 2319 = 41, 2519 = 43, 2919 = 47, 3219 = 59, 3419 = 61, 3A19 = 67, 3E19 = 71, 3G19 = 73, 4319 = 79, 4719 = 83, 4D19 = 89, 5219 = 97, 5819 = 103, 5C19 = 107, 5E19 = 109, 5I19 = 113, 7419 = 137, 7G19 = 149, 7I19 = 151, 8519 = 157, 8F19 = 167, 9219 = 173, 9A19 = 181, A119 = 191, A319 = 193, A919 = 199, BE19 = 223, BI19 = 227, C119 = 229, C519 = 233, D419 = 251, DG19 = 263, E319 = 269, E519 = 271, EB19 = 277, EH19 = 283, F819 = 293, G319 = 307, G719 = 311, GD19 = 317, HE19 = 337, I519 = 347, I719 = 349, IB19 = 353, 11319 = 383, 12219 = 401, 13D19 = 431, 14619 = 443, 14C19 = 449, 15519 = 461, 16419 = 479, 16C19 = 487, 18819 = 521, 1A619 = 557, 1AC19 = 563, 1AI19 = 569, 1CC19 = 601, 1D519 = 613, 1DB19 = 619, 1EG19 = 643, 1FF19 = 661, 1GC19 = 677, 1GI19 = 683, 1H719 = 691, 21A19 = 751, 22119 = 761, 22D19 = 773, 24B19 = 809, 27419 = 859, 27819 = 863, 28319 = 877, 29E19 = 907, 2A719 = 919, 2AH19 = 929, 2B619 = 937, 2BA19 = 941, 2BG19 = 947, 2CH19 = 967, 2D819 = 977, 2DE19 = 983, 2E919 = 997

20

1320 = 23, 1920 = 29, 3120 = 61, 3B20 = 71, 3D20 = 73, 3J20 = 79, 7B20 = 151, 7H20 = 157, 9120 = 181, 9B20 = 191, 9D20 = 193, 9H20 = 197, 9J20 = 199, B320 = 223, B720 = 227, B920 = 229, BD20 = 233, D320 = 263, D920 = 269, DB20 = 271, DH20 = 277, H720 = 347, H920 = 349, HD20 = 353, HJ20 = 359, J320 = 383, J920 = 389, JH20 = 397, 11B20 = 431, 11J20 = 439, 12H20 = 457, 13720 = 467, 14B20 = 491, 14J20 = 499, 15320 = 503, 15920 = 509, 16320 = 523, 18320 = 563, 18920 = 569, 18B20 = 571, 18H20 = 577, 19D20 = 593, 1A720 = 607, 1AH20 = 617, 1BB20 = 631, 1C720 = 647, 1CD20 = 653, 1CJ20 = 659, 1E320 = 683, 1FJ20 = 719, 1G720 = 727, 1GD20 = 733, 1J720 = 787

Bibliografia

  • Guy, Richard K.;  Woodrow, Robert E.;  The Lighter Side of Mathematics: Proceedings of the Eugene Strens Memorial Conference on Recreational Mathematics and Its History, Washington, Mathematical Association of America, 1994.

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