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Brillanti (numeri)

Rappresentazione dei numeri 

Peter Wallrodt definì “brillanti” i numeri che sono il prodotto di due primi, non necessariamente distinti, con ugual numero di cifre decimali. Includono quindi tutti i quadrati e vari semiprimi.

 

Solitamente ci si riferisce alla base 10, ma se il rapporto tra i due fattori è vicino a 1, questi avranno lo stesso numero di cifre in quasi tutte le basi, quindi un numero brillante resta tale in quasi tutte le basi.

In particolare i quadrati dei primi sono brillanti in qualsiasi base.

 

Gli unici pari sono i numeri della forma 2p, con p primo e inferiore alla base.

 

In base 10 quelli inferiori a 1000 sono: 4, 6, 9, 10, 14, 15, 21, 25, 35, 49, 121, 143, 169, 187, 209, 221, 247, 253, 289, 299, 319, 323, 341, 361, 377, 391, 403, 407, 437, 451, 473, 481, 493, 517, 527, 529, 533, 551, 559, 583, 589, 611, 629, 649, 667, 671, 689, 697, 703, 713, 731, 737, 767, 779, 781, 793, 799, 803, 817, 841, 851, 869, 871, 893, 899, 901, 913, 923, 943, 949, 961, 979, 989.

Qui trovate i numeri brillanti in base 10 sino a 107 (1.1Mbyte).

 

La tabella seguente riporta i numeri brillanti inferiori a 1000 nelle basi fino a 20.

Numero

Basi

4

2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20

6

2, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20

9

2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20

10

6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20

14

8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20

15

3, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20

21

3, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20

22

12, 13, 14, 15, 16, 17, 18, 19, 20

25

2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20

26

14, 15, 16, 17, 18, 19, 20

33

12, 13, 14, 15, 16, 17, 18, 19, 20

34

18, 19, 20

35

2, 3, 4, 5, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20

38

20

39

14, 15, 16, 17, 18, 19, 20

49

2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20

51

18, 19, 20

55

4, 5, 12, 13, 14, 15, 16, 17, 18, 19, 20

57

20

65

4, 5, 14, 15, 16, 17, 18, 19, 20

77

4, 5, 6, 7, 12, 13, 14, 15, 16, 17, 18, 19, 20

85

5, 18, 19, 20

91

4, 5, 6, 7, 14, 15, 16, 17, 18, 19, 20

95

5, 20

115

5

119

5, 6, 7, 18, 19, 20

121

2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20

133

5, 6, 7, 20

143

2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 14, 15, 16, 17, 18, 19, 20

161

5, 6, 7

169

2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20

187

3, 5, 6, 7, 8, 9, 10, 11, 18, 19, 20

203

6, 7

209

3, 5, 6, 7, 8, 9, 10, 11, 20

217

6, 7

221

3, 5, 6, 7, 8, 9, 10, 11, 12, 13, 18, 19, 20

247

3, 5, 6, 7, 8, 9, 10, 11, 12, 13, 20

253

3, 5, 6, 7, 8, 9, 10, 11

259

7

287

7

289

2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20

299

3, 5, 6, 7, 8, 9, 10, 11, 12, 13

301

7

319

6, 7, 8, 9, 10, 11

323

2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 20

329

7

341

6, 7, 8, 9, 10, 11

361

2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20

377

6, 7, 8, 9, 10, 11, 12, 13

391

2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17

403

6, 7, 8, 9, 10, 11, 12, 13

407

7, 8, 9, 10, 11

437

2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19

451

7, 8, 9, 10, 11

473

7, 8, 9, 10, 11

481

7, 8, 9, 10, 11, 12, 13

493

2, 4, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17

517

7, 8, 9, 10, 11

527

2, 4, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17

529

2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20

533

7, 8, 9, 10, 11, 12, 13

551

2, 4, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19

559

7, 8, 9, 10, 11, 12, 13

583

8, 9, 10, 11

589

2, 4, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19

611

7, 8, 9, 10, 11, 12, 13

629

4, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17

649

8, 9, 10, 11

667

2, 4, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20

671

8, 9, 10, 11

689

8, 9, 10, 11, 12, 13

697

4, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17

703

4, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19

713

2, 4, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20

731

4, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17

737

9, 10, 11

767

8, 9, 10, 11, 12, 13

779

4, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19

781

9, 10, 11

793

8, 9, 10, 11, 12, 13

799

4, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17

803

9, 10, 11

817

4, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19

841

2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20

851

4, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20

869

9, 10, 11

871

9, 10, 11, 12, 13

893

4, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19

899

2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20

901

4, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17

913

10, 11

923

9, 10, 11, 12, 13

943

4, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20

949

9, 10, 11, 12, 13

961

2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20

979

10, 11

989

4, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20

 

Richard Heylen stimò, sulla base di considerazioni euristiche, che la densità di numeri brillanti di n cifre in base b tenda a 2 * log(b) / n^2 per n dispari.

 

I numeri brillanti con molte cifre sono utilizzati per applicazioni crittografiche, nonché come banco di prova per gli algoritmi di fattorizzazione.

 

Il minimo quadrato magico formato da numeri brillanti senza divisori comuni è il seguente (a sinistra il quadrato, a destra la scomposizione dei numeri brillanti).

60997

122329

943

 

181 • 337

149 • 821

23 • 41

1369

61423

121477

 

37 • 37

239 • 257

331 • 367

121903

517

61849

 

139 • 877

11 • 47

127 • 487

 

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