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Pseudoprimi standard di Somer – Lucas

Sequenze  Teoria dei numeri 

Uno pseudoprimo n di Somer – Lucas si dice “standard” se Formula per la definizione degli pseudoprimi standard di Somer – Lucas, dove il prodotto va calcolato sui primi che dividono n e qui e nel seguito Simbolo di Jacobi(D, n) indica il simbolo di Jacobi.

Gli pseudoprimi di Somer – Lucas che non sono standard si dicono “eccezionali”.

 

Nel 2007 Walter Carlip e Lawrence Somer dimostrarono che:

  • per ogni valore di d = (n – (D | n)) / k gli pseudoprimi eccezionali di Somer – Lucas sono in numero finito;

  • ogni pseudoprimo standard di Somer – Lucas non multiplo di quadrati è un numero di Carmichael – Lucas;

  • gli pseudoprimi standard di Somer – Lucas con meno di 16 fattori primi non sono multipli di quadrati;

  • gli pseudoprimi standard di Somer – Lucas con d ≤ 984 non sono multipli di quadrati.

 

La tabella seguente mostra gli pseudoprimi eccezionali di Somer – Lucas fino a 1000 rispetto a P da 1 a 20 e Q da –20 a 20.

Pseudoprimo

(P, Q)

9

(2, -7), (2, 11), (4, -19), (4, -1), (4, 17), (8, -13), (8, 5), (10, -13), (14, -19), (14, -1), (14, 17), (16, -7), (16, 11), (18, -17), (18, -11), (18, -5), (18, 1), (18, 7), (18, 13), (18, 19), (20, -7), (20, 11)

15

(1, -7), (1, 8), (2, -13), (2, 17), (3, -8), (3, 7), (4, -7), (5, -19), (5, -16), (5, -4), (5, -1), (5, 11), (5, 14), (6, -17), (6, 13), (7, -13), (7, 2), (7, 17), (8, -13), (8, 17), (9, -17), (9, -2), (9, 13), (10, -19), (10, -1), (10, 11), (11, -7), (11, 8), (12, 7), (13, 2), (13, 17), (15, -14), (15, -11), (15, 1), (15, 4), (15, 16), (15, 19), (16, -7), (17, -13), (17, 2), (18, 7), (19, -7), (19, 8), (20, -19), (20, -1), (20, 11)

35

(1, -13), (1, -9), (1, 1), (1, 12), (2, -17), (2, -1), (2, 13), (3, -11), (4, -19), (4, 17), (5, -17), (5, -8), (5, -3), (5, 13), (5, 18), (6, -13), (6, 1), (7, -16), (7, -11), (7, -2), (7, -1), (7, 3), (7, 13), (7, 19), (8, 19), (9, -8), (9, 11), (10, 3), (10, 17), (11, -19), (11, -18), (11, -4), (11, 2), (11, 16), (11, 17), (12, -17), (12, -1), (12, 13), (13, -16), (13, -6), (13, -2), (13, 8), (13, 19), (14, -9), (14, 17), (15, -13), (15, -2), (15, 8), (16, -3), (16, 11), (17, -12), (17, -11), (17, 3), (17, 9), (18, -11), (19, -8), (19, -3), (19, 6), (19, 11), (20, -13)

143

(1, -15), (1, -7), (1, -4), (1, 1), (1, 6), (1, 7), (1, 19), (2, -9), (2, 5), (2, 15), (2, 17), (3, -5), (3, -2), (3, 8), (4, 5), (4, 7), (5, -19), (5, -9), (5, -1), (5, 2), (5, 7), (5, 12), (6, -1), (7, -20), (7, -3), (7, 5), (7, 8), (7, 18), (7, 19), (8, -19), (8, -1), (8, 17), (8, 19), (9, -5), (9, 5), (9, 16), (11, -8), (11, -2), (11, 4), (11, 5), (11, 15), (12, -7), (12, 1), (12, 7), (12, 19), (13, -17), (13, -16), (13, -12), (13, -9), (13, -1), (13, 4), (13, 10), (13, 16), (13, 17), (14, -19), (14, 9), (14, 19), (15, -2), (15, 2), (16, -15), (16, 3), (17, -19), (17, -12), (17, -10), (17, -8), (17, 1), (17, 3), (17, 7), (17, 14), (17, 18), (18, 7), (18, 19), (19, -16), (19, -14), (19, -8), (19, -5), (19, -3), (19, -1), (19, 6), (19, 8), (19, 10), (20, -7), (20, -1)

255

(3, 7), (5, -19), (5, -16), (5, 11), (6, 13), (7, -13), (9, -2), (10, -1), (11, -7), (11, 8), (13, 2), (15, -14), (15, 16), (15, 19), (17, -13), (17, 2), (19, -7), (20, -1), (20, 11)

323

(1, -18), (1, -15), (1, -11), (1, -1), (1, 1), (1, 7), (1, 8), (1, 18), (1, 20), (2, -5), (2, 11), (2, 13), (2, 15), (3, -16), (3, -13), (3, 1), (3, 10), (3, 20), (4, -7), (4, -3), (4, -1), (4, 7), (5, -12), (5, -9), (5, 7), (5, 13), (5, 14), (6, -11), (7, -6), (7, -3), (7, 1), (7, 11), (7, 13), (7, 20), (8, -7), (8, 5), (8, 9), (8, 11), (9, -11), (9, 2), (9, 5), (10, -3), (10, 1), (11, -20), (11, -14), (11, -1), (11, 12), (11, 15), (12, 13), (13, -16), (13, -8), (13, -7), (13, -2), (13, 11), (14, -13), (14, -9), (14, -5), (14, 5), (15, -13), (15, -2), (15, 7), (16, -9), (16, 1), (16, 3), (17, -10), (17, -6), (17, -5), (17, -3), (17, 3), (17, 11), (17, 14), (18, -11), (18, -1), (18, 1), (18, 7), (19, -11), (19, -9), (19, -6), (19, -5), (19, -4), (19, 8), (19, 12), (19, 13), (19, 15), (20, 1)

385

(1, 18), (2, 17), (3, -13), (3, -8), (4, 13), (5, -19), (5, -9), (5, -4), (6, 13), (7, -13), (7, -3), (7, 2), (8, -3), (8, 17), (10, 1), (10, 19), (11, -2), (12, 17), (13, -18), (13, 17), (15, -4), (16, 3), (17, -8), (17, 12), (18, -13), (19, -12), (19, -2), (19, 3), (20, -1)

899

(1, -17), (1, -10), (1, 1), (1, 3), (1, 4), (1, 7), (1, 11), (1, 14), (1, 19), (1, 20), (2, -17), (2, -13), (2, -9), (2, -7), (2, -1), (2, 13), (3, -8), (3, 5), (3, 13), (3, 16), (4, -15), (4, 7), (4, 15), (4, 19), (5, -19), (5, -18), (5, -11), (5, -4), (5, 7), (5, 9), (5, 13), (5, 17), (6, -5), (6, -1), (6, 1), (6, 7), (7, -16), (7, -13), (7, -12), (7, -5), (7, 2), (7, 13), (7, 15), (7, 16), (7, 19), (8, -3), (8, 19), (9, -19), (9, -13), (9, -8), (9, -5), (9, -4), (9, 10), (10, -7), (10, 7), (10, 11), (10, 17), (11, -18), (11, -9), (11, -3), (11, 2), (11, 5), (11, 6), (11, 13), (11, 20), (12, -11), (12, -5), (12, 17), (13, -20), (13, -9), (13, -6), (13, 3), (13, 9), (13, 14), (13, 17), (13, 18), (14, -17), (14, -1), (14, 9), (14, 17), (15, -17), (15, -7), (15, 1), (15, 8), (15, 17), (16, -5), (16, 17), (17, -20), (17, -14), (17, -10), (17, -7), (17, -1), (17, 4), (17, 9), (17, 10), (18, -17), (18, 5), (18, 11), (19, -15), (19, -14), (19, -2), (19, 4), (19, 8), (19, 11), (19, 17), (20, -13), (20, -1), (20, 3)

 

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