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Pseudoprimi di Eulero – Lucas

Sequenze  Teoria dei numeri 

Data una successione di Lucas generalizzata Un generata da P e Q, con P e Q interi non nulli e P2 diverso da 4Q, e la corrispondente successione associata Vn (v. pseudoprimi di Lucas), se n è primo e non ha divisori comuni col prodotto DQ, allora, indicando con Simbolo di Jacobi (Q | n) il simbolo di Jacobi:

  • Congruenza soddisfatta da n, se Simbolo di Jacobi (Q | n) = 1;

  • Congruenza soddisfatta da n, se Simbolo di Jacobi (Q | n) = –1.

Un numero composto dispari che soddisfi la stessa condizione si dice “pseudoprimo di Eulero – Lucas” rispetto a P e Q.

Tutti gli pseudoprimi di Eulero – Lucas che non hanno divisori comuni con Q sono anche pseudoprimi di Lucas rispetto a P e Q.

 

Se n è uno pseudoprimo di Lucas rispetto a P e Q e uno pseudoprimo di Eulero – Jacobi in base Q, non ha divisori in comune con DQ e Congruenza soddisfatta da n, allora n è pseudoprimo di Eulero – Lucas rispetto agli stessi P e Q (Robert Baillie e Samuel S. Wagstaff Jr., 1980).

 

Se n è pseudoprimo di Eulero – Lucas rispetto a P e QSimbolo di Jacobi (Q | n) = –1 oppure Congruenza soddisfatta da n, allora n è pseudoprimo forte di Lucas rispetto agli stessi P e Q (Robert Baillie e Samuel S. Wagstaff Jr., 1980).

 

Se n è pseudoprimo di Eulero – Lucas rispetto a P e Q e non ha divisori comuni con P, allora è pseudoprimo di Lucas, pseudoprimo di Lucas di seconda specie, pseudoprimo di Dicksonpseudoprimo di Dickson di seconda specie rispetto a P e Q (Andrzej Rotkiewicz, 2000).

 

Se n è pseudoprimo di Eulero – Lucas rispetto a P e Q, non ha divisori comuni con DPQ, ed è pseudoprimo di Dickson rispetto a P e Q, allora è pseudoprimo di Eulero – Jacobi in base Q (Andrzej Rotkiewicz, 2000).

 

Se n non è multiplo di quadrati, è pseudoprimo di Dickson di seconda specie rispetto a P e Q ed è pseudoprimo di Eulero – Jacobi in base Q, allora è pseudoprimo di Eulero – Lucas rispetto a P e Q (Andrzej Rotkiewicz, 2000).

 

E.A. Parberry dimostrò nel 1970 che esistono infiniti pseudoprimi di Eulero – Lucas rispetto a 1 e –1, cioè rispetto alle sequenze di Fibonacci e di Lucas; quelli inferiori a 106 sono: 4181, 5777, 6721, 10877, 13201, 34561, 51841, 64681, 67861, 75077, 90061, 96049, 97921, 113573, 118441, 162133, 163081, 186961, 197209, 219781, 252601, 254321, 257761, 268801, 283361, 302101, 303101, 330929, 399001, 433621, 489601, 512461, 520801, 556421, 636641, 638189, 655201, 722261, 852841, 853469, 925681, 999941.

Qui trovate gli pseudoprimi di Eulero – Lucas rispetto a 1 e –1inferiori a 109 (M. Fiorentini, 2017).

 

La tabella seguente mostra gli pseudoprimi di Eulero – Lucas fino a 100 rispetto a P da 1 a 100 e Q da –100 a 100.

Pseudoprimo

(P, Q)

9

(1, –4), (1, +5), (2, –7), (2, +2), (4, –1), (4, +8), (5, –1), (5, +8), (7, –7), (7, +2), (8, –4), (8, +5)

15

(1, –7), (1, +8), (2, –13), (2, +2), (4, –7), (4, +8), (7, –13), (7, +2), (8, –13), (8, +2), (11, –7), (11, +8), (13, –13), (13, +2), (14, –7), (14, +8)

25

(1, –24), (1, –12), (1, –8), (1, +1), (1, +13), (1, +17), (2, –23), (2, –21), (2, –7), (2, +2), (2, +4), (2, +18), (3, –22), (3, –16), (3, –8), (3, +3), (3, +9), (3, +17), (4, –17), (4, –9), (4, –3), (4, +8), (4, +16), (4, +22), (6, –14), (6, –13), (6, –7), (6, +11), (6, +12), (6, +18), (7, –17), (7, –13), (7, –1), (7, +8), (7, +12), (7, +24), (8, –18), (8, –12), (8, –11), (8, +7), (8, +13), (8, +14), (9, –23), (9, –22), (9, –19), (9, +2), (9, +3), (9, +6), (11, –18), (11, –4), (11, –2), (11, +7), (11, +21), (11, +23), (12, –6), (12, –3), (12, –2), (12, +19), (12, +22), (12, +23), (13, –6), (13, –3), (13, –2), (13, +19), (13, +22), (13, +23), (14, –18), (14, –4), (14, –2), (14, +7), (14, +21), (14, +23), (16, –23), (16, –22), (16, –19), (16, +2), (16, +3), (16, +6), (17, –18), (17, –12), (17, –11), (17, +7), (17, +13), (17, +14), (18, –17), (18, –13), (18, –1), (18, +8), (18, +12), (18, +24), (19, –14), (19, –13), (19, –7), (19, +11), (19, +12), (19, +18), (21, –17), (21, –9), (21, –3), (21, +8), (21, +16), (21, +22), (22, –22), (22, –16), (22, –8), (22, +3), (22, +9), (22, +17), (23, –23), (23, –21), (23, –7), (23, +2), (23, +4), (23, +18), (24, –24), (24, –12), (24, –8), (24, +1), (24, +13), (24, +17)

33

(1, –16), (1, +17), (2, –31), (2, +2), (3, –23), (3, +10), (4, –25), (4, +8), (5, –4), (5, +29), (7, –25), (7, +8), (8, –1), (8, +32), (10, –16), (10, +17), (11, –19), (11, +14), (13, –31), (13, +2), (14, –1), (14, +32), (16, –4), (16, +29), (17, –4), (17, +29), (19, –1), (19, +32), (20, –31), (20, +2), (22, –19), (22, +14), (23, –16), (23, +17), (25, –1), (25, +32), (26, –25), (26, +8), (28, –4), (28, +29), (29, –25), (29, +8), (30, –23), (30, +10), (31, –31), (31, +2), (32, –16), (32, +17)

35

(1, –34), (1, –23), (1, –9), (1, +1), (1, +12), (1, +26), (2, –31), (2, –22), (2, +4), (2, +13), (3, –32), (3, –26), (3, +3), (3, +9), (4, –19), (4, –18), (4, +16), (4, +17), (5, –22), (5, –8), (5, +13), (5, +27), (6, –34), (6, –23), (6, –9), (6, +1), (6, +12), (6, +26), (7, –32), (7, –22), (7, –11), (7, –2), (7, +3), (7, +13), (7, +24), (7, +33), (8, –16), (8, –6), (8, –2), (8, +19), (8, +29), (8, +33), (9, –24), (9, –8), (9, +11), (9, +27), (10, –32), (10, –18), (10, +3), (10, +17), (11, –19), (11, –18), (11, +16), (11, +17), (12, –31), (12, –22), (12, +4), (12, +13), (13, –16), (13, –6), (13, –2), (13, +19), (13, +29), (13, +33), (14, –23), (14, –18), (14, –8), (14, –4), (14, +12), (14, +17), (14, +27), (14, +31), (15, –23), (15, –2), (15, +12), (15, +33), (16, –24), (16, –8), (16, +11), (16, +27), (17, –32), (17, –26), (17, +3), (17, +9), (18, –32), (18, –26), (18, +3), (18, +9), (19, –24), (19, –8), (19, +11), (19, +27), (20, –23), (20, –2), (20, +12), (20, +33), (21, –23), (21, –18), (21, –8), (21, –4), (21, +12), (21, +17), (21, +27), (21, +31), (22, –16), (22, –6), (22, –2), (22, +19), (22, +29), (22, +33), (23, –31), (23, –22), (23, +4), (23, +13), (24, –19), (24, –18), (24, +16), (24, +17), (25, –32), (25, –18), (25, +3), (25, +17), (26, –24), (26, –8), (26, +11), (26, +27), (27, –16), (27, –6), (27, –2), (27, +19), (27, +29), (27, +33), (28, –32), (28, –22), (28, –11), (28, –2), (28, +3), (28, +13), (28, +24), (28, +33), (29, –34), (29, –23), (29, –9), (29, +1), (29, +12), (29, +26), (30, –22), (30, –8), (30, +13), (30, +27), (31, –19), (31, –18), (31, +16), (31, +17), (32, –32), (32, –26), (32, +3), (32, +9), (33, –31), (33, –22), (33, +4), (33, +13), (34, –34), (34, –23), (34, –9), (34, +1), (34, +12), (34, +26)

39

(1, –19), (1, +20), (2, –37), (2, +2), (4, –31), (4, +8), (5, –7), (5, +32), (7, –34), (7, +5), (8, –7), (8, +32), (10, –28), (10, +11), (11, –37), (11, +2), (14, –19), (14, +20), (16, –28), (16, +11), (17, –31), (17, +8), (19, –34), (19, +5), (20, –34), (20, +5), (22, –31), (22, +8), (23, –28), (23, +11), (25, –19), (25, +20), (28, –37), (28, +2), (29, –28), (29, +11), (31, –7), (31, +32), (32, –34), (32, +5), (34, –7), (34, +32), (35, –31), (35, +8), (37, –37), (37, +2), (38, –19), (38, +20)

49

(1, –48), (1, –43), (1, –24), (1, –16), (1, –4), (1, +1), (1, +6), (1, +25), (1, +33), (1, +45), (2, –47), (2, –45), (2, –25), (2, –16), (2, –15), (2, +2), (2, +4), (2, +24), (2, +33), (2, +34), (3, –46), (3, –44), (3, –40), (3, –36), (3, –20), (3, +3), (3, +5), (3, +9), (3, +13), (3, +29), (4, –41), (4, –33), (4, –15), (4, –11), (4, –2), (4, +8), (4, +16), (4, +34), (4, +38), (4, +47), (5, –46), (5, –24), (5, –12), (5, –8), (5, –2), (5, +3), (5, +25), (5, +37), (5, +41), (5, +47), (6, –46), (6, –37), (6, –31), (6, –29), (6, –13), (6, +3), (6, +12), (6, +18), (6, +20), (6, +36), (8, –44), (8, –34), (8, –17), (8, –11), (8, –8), (8, +5), (8, +15), (8, +32), (8, +38), (8, +41), (9, –33), (9, –30), (9, –22), (9, –17), (9, –4), (9, +16), (9, +19), (9, +27), (9, +32), (9, +45), (10, –48), (10, –47), (10, –37), (10, –32), (10, –8), (10, +1), (10, +2), (10, +12), (10, +17), (10, +41), (11, –43), (11, –26), (11, –25), (11, –13), (11, –9), (11, +6), (11, +23), (11, +24), (11, +36), (11, +40), (12, –37), (12, –26), (12, –18), (12, –3), (12, –1), (12, +12), (12, +23), (12, +31), (12, +46), (12, +48), (13, –39), (13, –38), (13, –27), (13, –15), (13, –9), (13, +10), (13, +11), (13, +22), (13, +34), (13, +40), (15, –23), (15, –22), (15, –20), (15, –18), (15, –10), (15, +26), (15, +27), (15, +29), (15, +31), (15, +39), (16, –44), (16, –38), (16, –32), (16, –29), (16, –19), (16, +5), (16, +11), (16, +17), (16, +20), (16, +30), (17, –30), (17, –29), (17, –27), (17, –18), (17, –5), (17, +19), (17, +20), (17, +22), (17, +31), (17, +44), (18, –39), (18, –34), (18, –22), (18, –19), (18, –16), (18, +10), (18, +15), (18, +27), (18, +30), (18, +33), (19, –43), (19, –40), (19, –39), (19, –31), (19, –23), (19, +6), (19, +9), (19, +10), (19, +18), (19, +26), (20, –45), (20, –41), (20, –32), (20, –30), (20, –1), (20, +4), (20, +8), (20, +17), (20, +19), (20, +48), (22, –36), (22, –25), (22, –6), (22, –3), (22, –2), (22, +13), (22, +24), (22, +43), (22, +46), (22, +47), (23, –36), (23, –11), (23, –10), (23, –9), (23, –5), (23, +13), (23, +38), (23, +39), (23, +40), (23, +44), (24, –23), (24, –12), (24, –6), (24, –4), (24, –1), (24, +26), (24, +37), (24, +43), (24, +45), (24, +48), (25, –23), (25, –12), (25, –6), (25, –4), (25, –1), (25, +26), (25, +37), (25, +43), (25, +45), (25, +48), (26, –36), (26, –11), (26, –10), (26, –9), (26, –5), (26, +13), (26, +38), (26, +39), (26, +40), (26, +44), (27, –36), (27, –25), (27, –6), (27, –3), (27, –2), (27, +13), (27, +24), (27, +43), (27, +46), (27, +47), (29, –45), (29, –41), (29, –32), (29, –30), (29, –1), (29, +4), (29, +8), (29, +17), (29, +19), (29, +48), (30, –43), (30, –40), (30, –39), (30, –31), (30, –23), (30, +6), (30, +9), (30, +10), (30, +18), (30, +26), (31, –39), (31, –34), (31, –22), (31, –19), (31, –16), (31, +10), (31, +15), (31, +27), (31, +30), (31, +33), (32, –30), (32, –29), (32, –27), (32, –18), (32, –5), (32, +19), (32, +20), (32, +22), (32, +31), (32, +44), (33, –44), (33, –38), (33, –32), (33, –29), (33, –19), (33, +5), (33, +11), (33, +17), (33, +20), (33, +30), (34, –23), (34, –22), (34, –20), (34, –18), (34, –10), (34, +26), (34, +27), (34, +29), (34, +31), (34, +39), (36, –39), (36, –38), (36, –27), (36, –15), (36, –9), (36, +10), (36, +11), (36, +22), (36, +34), (36, +40), (37, –37), (37, –26), (37, –18), (37, –3), (37, –1), (37, +12), (37, +23), (37, +31), (37, +46), (37, +48), (38, –43), (38, –26), (38, –25), (38, –13), (38, –9), (38, +6), (38, +23), (38, +24), (38, +36), (38, +40), (39, –48), (39, –47), (39, –37), (39, –32), (39, –8), (39, +1), (39, +2), (39, +12), (39, +17), (39, +41), (40, –33), (40, –30), (40, –22), (40, –17), (40, –4), (40, +16), (40, +19), (40, +27), (40, +32), (40, +45), (41, –44), (41, –34), (41, –17), (41, –11), (41, –8), (41, +5), (41, +15), (41, +32), (41, +38), (41, +41), (43, –46), (43, –37), (43, –31), (43, –29), (43, –13), (43, +3), (43, +12), (43, +18), (43, +20), (43, +36), (44, –46), (44, –24), (44, –12), (44, –8), (44, –2), (44, +3), (44, +25), (44, +37), (44, +41), (44, +47), (45, –41), (45, –33), (45, –15), (45, –11), (45, –2), (45, +8), (45, +16), (45, +34), (45, +38), (45, +47), (46, –46), (46, –44), (46, –40), (46, –36), (46, –20), (46, +3), (46, +5), (46, +9), (46, +13), (46, +29), (47, –47), (47, –45), (47, –25), (47, –16), (47, –15), (47, +2), (47, +4), (47, +24), (47, +33), (47, +34), (48, –48), (48, –43), (48, –24), (48, –16), (48, –4), (48, +1), (48, +6), (48, +25), (48, +33), (48, +45)

55

(1, –54), (1, –27), (1, +1), (1, +28), (2, –53), (2, –51), (2, +2), (2, +4), (3, –46), (3, –23), (3, +9), (3, +32), (4, –47), (4, –39), (4, +8), (4, +16), (6, –37), (6, –19), (6, +18), (6, +36), (7, –6), (7, –3), (7, +49), (7, +52), (8, –46), (8, –23), (8, +9), (8, +32), (9, –42), (9, –29), (9, +13), (9, +26), (12, –38), (12, –21), (12, +17), (12, +34), (13, –53), (13, –51), (13, +2), (13, +4), (14, –24), (14, –12), (14, +31), (14, +43), (16, –37), (16, –19), (16, +18), (16, +36), (17, –48), (17, –41), (17, +7), (17, +14), (18, –6), (18, –3), (18, +49), (18, +52), (19, –24), (19, –12), (19, +31), (19, +43), (21, –54), (21, –27), (21, +1), (21, +28), (23, –38), (23, –21), (23, +17), (23, +34), (24, –42), (24, –29), (24, +13), (24, +26), (26, –47), (26, –39), (26, +8), (26, +16), (27, –48), (27, –41), (27, +7), (27, +14), (28, –48), (28, –41), (28, +7), (28, +14), (29, –47), (29, –39), (29, +8), (29, +16), (31, –42), (31, –29), (31, +13), (31, +26), (32, –38), (32, –21), (32, +17), (32, +34), (34, –54), (34, –27), (34, +1), (34, +28), (36, –24), (36, –12), (36, +31), (36, +43), (37, –6), (37, –3), (37, +49), (37, +52), (38, –48), (38, –41), (38, +7), (38, +14), (39, –37), (39, –19), (39, +18), (39, +36), (41, –24), (41, –12), (41, +31), (41, +43), (42, –53), (42, –51), (42, +2), (42, +4), (43, –38), (43, –21), (43, +17), (43, +34), (46, –42), (46, –29), (46, +13), (46, +26), (47, –46), (47, –23), (47, +9), (47, +32), (48, –6), (48, –3), (48, +49), (48, +52), (49, –37), (49, –19), (49, +18), (49, +36), (51, –47), (51, –39), (51, +8), (51, +16), (52, –46), (52, –23), (52, +9), (52, +32), (53, –53), (53, –51), (53, +2), (53, +4), (54, –54), (54, –27), (54, +1), (54, +28)

57

(1, –28), (1, +29), (2, –55), (2, +2), (4, –49), (4, +8), (5, –16), (5, +41), (7, –4), (7, +53), (8, –25), (8, +32), (10, –7), (10, +50), (11, –25), (11, +32), (13, –1), (13, +56), (14, –16), (14, +41), (16, –43), (16, +14), (17, –55), (17, +2), (20, –28), (20, +29), (22, –43), (22, +14), (23, –49), (23, +8), (25, –1), (25, +56), (26, –4), (26, +53), (28, –7), (28, +50), (29, –7), (29, +50), (31, –4), (31, +53), (32, –1), (32, +56), (34, –49), (34, +8), (35, –43), (35, +14), (37, –28), (37, +29), (40, –55), (40, +2), (41, –43), (41, +14), (43, –16), (43, +41), (44, –1), (44, +56), (46, –25), (46, +32), (47, –7), (47, +50), (49, –25), (49, +32), (50, –4), (50, +53), (52, –16), (52, +41), (53, –49), (53, +8), (55, –55), (55, +2), (56, –28), (56, +29)

63

(1, –31), (1, +32), (2, –61), (2, +2), (4, –55), (4, +8), (5, –19), (5, +44), (7, –61), (7, –52), (7, –34), (7, +2), (7, +11), (7, +29), (8, –31), (8, +32), (9, –61), (9, –47), (9, –40), (9, –26), (9, –19), (9, –5), (9, +2), (9, +16), (9, +23), (9, +37), (9, +44), (9, +58), (10, –13), (10, +50), (11, –34), (11, +29), (13, –10), (13, +53), (14, –55), (14, –19), (14, –10), (14, +8), (14, +44), (14, +53), (16, –61), (16, +2), (17, –13), (17, +50), (18, –62), (18, –55), (18, –41), (18, –34), (18, –20), (18, –13), (18, +1), (18, +8), (18, +22), (18, +29), (18, +43), (18, +50), (19, –40), (19, +23), (20, –52), (20, +11), (22, –10), (22, +53), (23, –19), (23, +44), (25, –34), (25, +29), (26, –40), (26, +23), (27, –59), (27, –52), (27, –38), (27, –31), (27, –17), (27, –10), (27, +4), (27, +11), (27, +25), (27, +32), (27, +46), (27, +53), (28, –40), (28, –31), (28, –13), (28, +23), (28, +32), (28, +50), (29, –52), (29, +11), (31, –55), (31, +8), (32, –55), (32, +8), (34, –52), (34, +11), (35, –40), (35, –31), (35, –13), (35, +23), (35, +32), (35, +50), (36, –59), (36, –52), (36, –38), (36, –31), (36, –17), (36, –10), (36, +4), (36, +11), (36, +25), (36, +32), (36, +46), (36, +53), (37, –40), (37, +23), (38, –34), (38, +29), (40, –19), (40, +44), (41, –10), (41, +53), (43, –52), (43, +11), (44, –40), (44, +23), (45, –62), (45, –55), (45, –41), (45, –34), (45, –20), (45, –13), (45, +1), (45, +8), (45, +22), (45, +29), (45, +43), (45, +50), (46, –13), (46, +50), (47, –61), (47, +2), (49, –55), (49, –19), (49, –10), (49, +8), (49, +44), (49, +53), (50, –10), (50, +53), (52, –34), (52, +29), (53, –13), (53, +50), (54, –61), (54, –47), (54, –40), (54, –26), (54, –19), (54, –5), (54, +2), (54, +16), (54, +23), (54, +37), (54, +44), (54, +58), (55, –31), (55, +32), (56, –61), (56, –52), (56, –34), (56, +2), (56, +11), (56, +29), (58, –19), (58, +44), (59, –55), (59, +8), (61, –61), (61, +2), (62, –31), (62, +32)

65

(1, –64), (1, –32), (1, +1), (1, +33), (2, –63), (2, –61), (2, +2), (2, +4), (3, –56), (3, –28), (3, +9), (3, +37), (4, –57), (4, –49), (4, +8), (4, +16), (6, –47), (6, –29), (6, +18), (6, +36), (7, –16), (7, –8), (7, +49), (7, +57), (8, –33), (8, –1), (8, +32), (8, +64), (9, –57), (9, –49), (9, +8), (9, +16), (11, –37), (11, –9), (11, +28), (11, +56), (12, –58), (12, –51), (12, +7), (12, +14), (14, –64), (14, –32), (14, +1), (14, +33), (16, –4), (16, –2), (16, +61), (16, +63), (17, –36), (17, –18), (17, +29), (17, +47), (18, –33), (18, –1), (18, +32), (18, +64), (19, –47), (19, –29), (19, +18), (19, +36), (21, –14), (21, –7), (21, +51), (21, +58), (22, –36), (22, –18), (22, +29), (22, +47), (23, –56), (23, –28), (23, +9), (23, +37), (24, –37), (24, –9), (24, +28), (24, +56), (27, –58), (27, –51), (27, +7), (27, +14), (28, –63), (28, –61), (28, +2), (28, +4), (29, –4), (29, –2), (29, +61), (29, +63), (31, –14), (31, –7), (31, +51), (31, +58), (32, –16), (32, –8), (32, +49), (32, +57), (33, –16), (33, –8), (33, +49), (33, +57), (34, –14), (34, –7), (34, +51), (34, +58), (36, –4), (36, –2), (36, +61), (36, +63), (37, –63), (37, –61), (37, +2), (37, +4), (38, –58), (38, –51), (38, +7), (38, +14), (41, –37), (41, –9), (41, +28), (41, +56), (42, –56), (42, –28), (42, +9), (42, +37), (43, –36), (43, –18), (43, +29), (43, +47), (44, –14), (44, –7), (44, +51), (44, +58), (46, –47), (46, –29), (46, +18), (46, +36), (47, –33), (47, –1), (47, +32), (47, +64), (48, –36), (48, –18), (48, +29), (48, +47), (49, –4), (49, –2), (49, +61), (49, +63), (51, –64), (51, –32), (51, +1), (51, +33), (53, –58), (53, –51), (53, +7), (53, +14), (54, –37), (54, –9), (54, +28), (54, +56), (56, –57), (56, –49), (56, +8), (56, +16), (57, –33), (57, –1), (57, +32), (57, +64), (58, –16), (58, –8), (58, +49), (58, +57), (59, –47), (59, –29), (59, +18), (59, +36), (61, –57), (61, –49), (61, +8), (61, +16), (62, –56), (62, –28), (62, +9), (62, +37), (63, –63), (63, –61), (63, +2), (63, +4), (64, –64), (64, –32), (64, +1), (64, +33)

77

(1, –76), (1, +1), (2, –73), (2, +4), (3, –68), (3, +9), (4, –61), (4, +16), (5, –52), (5, +25), (6, –41), (6, +36), (8, –13), (8, +64), (9, –73), (9, +4), (10, –54), (10, +23), (12, –10), (12, +67), (13, –62), (13, +15), (15, –6), (15, +71), (16, –52), (16, +25), (17, –19), (17, +58), (18, –61), (18, +16), (19, –24), (19, +53), (20, –62), (20, +15), (23, –10), (23, +67), (24, –40), (24, +37), (25, –68), (25, +9), (26, –17), (26, +60), (27, –41), (27, +36), (29, –6), (29, +71), (30, –24), (30, +53), (31, –40), (31, +37), (32, –54), (32, +23), (34, –76), (34, +1), (36, –13), (36, +64), (37, –17), (37, +60), (38, –19), (38, +58), (39, –19), (39, +58), (40, –17), (40, +60), (41, –13), (41, +64), (43, –76), (43, +1), (45, –54), (45, +23), (46, –40), (46, +37), (47, –24), (47, +53), (48, –6), (48, +71), (50, –41), (50, +36), (51, –17), (51, +60), (52, –68), (52, +9), (53, –40), (53, +37), (54, –10), (54, +67), (57, –62), (57, +15), (58, –24), (58, +53), (59, –61), (59, +16), (60, –19), (60, +58), (61, –52), (61, +25), (62, –6), (62, +71), (64, –62), (64, +15), (65, –10), (65, +67), (67, –54), (67, +23), (68, –73), (68, +4), (69, –13), (69, +64), (71, –41), (71, +36), (72, –52), (72, +25), (73, –61), (73, +16), (74, –68), (74, +9), (75, –73), (75, +4), (76, –76), (76, +1)

81

(1, –40), (1, +41), (2, –79), (2, +2), (4, –73), (4, +8), (5, –28), (5, +53), (7, –16), (7, +65), (8, –49), (8, +32), (10, –31), (10, +50), (11, –61), (11, +20), (13, –37), (13, +44), (14, –64), (14, +17), (16, –34), (16, +47), (17, –58), (17, +23), (19, –22), (19, +59), (20, –43), (20, +38), (22, –1), (22, +80), (23, –19), (23, +62), (25, –52), (25, +29), (26, –67), (26, +14), (28, –13), (28, +68), (29, –25), (29, +56), (31, –46), (31, +35), (32, –55), (32, +26), (34, –70), (34, +11), (35, –76), (35, +5), (37, –4), (37, +77), (38, –7), (38, +74), (40, –10), (40, +71), (41, –10), (41, +71), (43, –7), (43, +74), (44, –4), (44, +77), (46, –76), (46, +5), (47, –70), (47, +11), (49, –55), (49, +26), (50, –46), (50, +35), (52, –25), (52, +56), (53, –13), (53, +68), (55, –67), (55, +14), (56, –52), (56, +29), (58, –19), (58, +62), (59, –1), (59, +80), (61, –43), (61, +38), (62, –22), (62, +59), (64, –58), (64, +23), (65, –34), (65, +47), (67, –64), (67, +17), (68, –37), (68, +44), (70, –61), (70, +20), (71, –31), (71, +50), (73, –49), (73, +32), (74, –16), (74, +65), (76, –28), (76, +53), (77, –73), (77, +8), (79, –79), (79, +2), (80, –40), (80, +41)

85

(1, –84), (1, –28), (1, +1), (1, +57), (2, –81), (2, –27), (2, +4), (2, +58), (3, –82), (3, –76), (3, +3), (3, +9), (4, –69), (4, –23), (4, +16), (4, +62), (5, –37), (5, –3), (5, +48), (5, +82), (6, –73), (6, –49), (6, +12), (6, +36), (7, –36), (7, –12), (7, +49), (7, +73), (8, –21), (8, –7), (8, +64), (8, +78), (9, –58), (9, –4), (9, +27), (9, +81), (10, –63), (10, –12), (10, +22), (10, +73), (11, –73), (11, –49), (11, +12), (11, +36), (12, –37), (12, –26), (12, +48), (12, +59), (13, –57), (13, –1), (13, +28), (13, +84), (14, –59), (14, –48), (14, +26), (14, +37), (15, –78), (15, –27), (15, +7), (15, +58), (16, –84), (16, –28), (16, +1), (16, +57), (17, –82), (17, –62), (17, –57), (17, –37), (17, –27), (17, –22), (17, –12), (17, –7), (17, +3), (17, +23), (17, +28), (17, +48), (17, +58), (17, +63), (17, +73), (17, +78), (18, –62), (18, –16), (18, +23), (18, +69), (19, –78), (19, –64), (19, +7), (19, +21), (20, –82), (20, –48), (20, +3), (20, +37), (21, –69), (21, –23), (21, +16), (21, +62), (22, –37), (22, –26), (22, +48), (22, +59), (23, –66), (23, –22), (23, +19), (23, +63), (24, –63), (24, –19), (24, +22), (24, +66), (25, –58), (25, –7), (25, +27), (25, +78), (26, –58), (26, –4), (26, +27), (26, +81), (27, –36), (27, –12), (27, +49), (27, +73), (28, –66), (28, –22), (28, +19), (28, +63), (29, –9), (29, –3), (29, +76), (29, +82), (30, –57), (30, –23), (30, +28), (30, +62), (31, –59), (31, –48), (31, +26), (31, +37), (32, –81), (32, –27), (32, +4), (32, +58), (33, –62), (33, –16), (33, +23), (33, +69), (34, –78), (34, –73), (34, –63), (34, –58), (34, –48), (34, –28), (34, –23), (34, –3), (34, +7), (34, +12), (34, +22), (34, +27), (34, +37), (34, +57), (34, +62), (34, +82), (35, –62), (35, –28), (35, +23), (35, +57), (36, –78), (36, –64), (36, +7), (36, +21), (37, –82), (37, –76), (37, +3), (37, +9), (38, –57), (38, –1), (38, +28), (38, +84), (39, –9), (39, –3), (39, +76), (39, +82), (40, –73), (40, –22), (40, +12), (40, +63), (41, –63), (41, –19), (41, +22), (41, +66), (42, –21), (42, –7), (42, +64), (42, +78), (43, –21), (43, –7), (43, +64), (43, +78), (44, –63), (44, –19), (44, +22), (44, +66), (45, –73), (45, –22), (45, +12), (45, +63), (46, –9), (46, –3), (46, +76), (46, +82), (47, –57), (47, –1), (47, +28), (47, +84), (48, –82), (48, –76), (48, +3), (48, +9), (49, –78), (49, –64), (49, +7), (49, +21), (50, –62), (50, –28), (50, +23), (50, +57), (51, –78), (51, –73), (51, –63), (51, –58), (51, –48), (51, –28), (51, –23), (51, –3), (51, +7), (51, +12), (51, +22), (51, +27), (51, +37), (51, +57), (51, +62), (51, +82), (52, –62), (52, –16), (52, +23), (52, +69), (53, –81), (53, –27), (53, +4), (53, +58), (54, –59), (54, –48), (54, +26), (54, +37), (55, –57), (55, –23), (55, +28), (55, +62), (56, –9), (56, –3), (56, +76), (56, +82), (57, –66), (57, –22), (57, +19), (57, +63), (58, –36), (58, –12), (58, +49), (58, +73), (59, –58), (59, –4), (59, +27), (59, +81), (60, –58), (60, –7), (60, +27), (60, +78), (61, –63), (61, –19), (61, +22), (61, +66), (62, –66), (62, –22), (62, +19), (62, +63), (63, –37), (63, –26), (63, +48), (63, +59), (64, –69), (64, –23), (64, +16), (64, +62), (65, –82), (65, –48), (65, +3), (65, +37), (66, –78), (66, –64), (66, +7), (66, +21), (67, –62), (67, –16), (67, +23), (67, +69), (68, –82), (68, –62), (68, –57), (68, –37), (68, –27), (68, –22), (68, –12), (68, –7), (68, +3), (68, +23), (68, +28), (68, +48), (68, +58), (68, +63), (68, +73), (68, +78), (69, –84), (69, –28), (69, +1), (69, +57), (70, –78), (70, –27), (70, +7), (70, +58), (71, –59), (71, –48), (71, +26), (71, +37), (72, –57), (72, –1), (72, +28), (72, +84), (73, –37), (73, –26), (73, +48), (73, +59), (74, –73), (74, –49), (74, +12), (74, +36), (75, –63), (75, –12), (75, +22), (75, +73), (76, –58), (76, –4), (76, +27), (76, +81), (77, –21), (77, –7), (77, +64), (77, +78), (78, –36), (78, –12), (78, +49), (78, +73), (79, –73), (79, –49), (79, +12), (79, +36), (80, –37), (80, –3), (80, +48), (80, +82), (81, –69), (81, –23), (81, +16), (81, +62), (82, –82), (82, –76), (82, +3), (82, +9), (83, –81), (83, –27), (83, +4), (83, +58), (84, –84), (84, –28), (84, +1), (84, +57)

87

(1, –43), (1, +44), (2, –85), (2, +2), (4, –79), (4, +8), (5, –31), (5, +56), (7, –19), (7, +68), (8, –55), (8, +32), (10, –37), (10, +50), (11, –70), (11, +17), (13, –46), (13, +41), (14, –76), (14, +11), (16, –46), (16, +41), (17, –73), (17, +14), (19, –37), (19, +50), (20, –61), (20, +26), (22, –19), (22, +68), (23, –40), (23, +47), (25, –79), (25, +8), (26, –10), (26, +77), (28, –43), (28, +44), (31, –85), (31, +2), (32, –10), (32, +77), (34, –31), (34, +56), (35, –40), (35, +47), (37, –55), (37, +32), (38, –61), (38, +26), (40, –70), (40, +17), (41, –73), (41, +14), (43, –76), (43, +11), (44, –76), (44, +11), (46, –73), (46, +14), (47, –70), (47, +17), (49, –61), (49, +26), (50, –55), (50, +32), (52, –40), (52, +47), (53, –31), (53, +56), (55, –10), (55, +77), (56, –85), (56, +2), (59, –43), (59, +44), (61, –10), (61, +77), (62, –79), (62, +8), (64, –40), (64, +47), (65, –19), (65, +68), (67, –61), (67, +26), (68, –37), (68, +50), (70, –73), (70, +14), (71, –46), (71, +41), (73, –76), (73, +11), (74, –46), (74, +41), (76, –70), (76, +17), (77, –37), (77, +50), (79, –55), (79, +32), (80, –19), (80, +68), (82, –31), (82, +56), (83, –79), (83, +8), (85, –85), (85, +2), (86, –43), (86, +44)

91

(1, –90), (1, +1), (2, –87), (2, +4), (3, –82), (3, +9), (4, –75), (4, +16), (5, –66), (5, +25), (6, –55), (6, +36), (8, –27), (8, +64), (9, –10), (9, +81), (10, –82), (10, +9), (11, –61), (11, +30), (12, –38), (12, +53), (15, –48), (15, +43), (16, –17), (16, +74), (17, –75), (17, +16), (18, –40), (18, +51), (19, –3), (19, +88), (20, –55), (20, +36), (22, –62), (22, +29), (23, –17), (23, +74), (24, –61), (24, +30), (25, –12), (25, +79), (27, –90), (27, +1), (29, –69), (29, +22), (30, –10), (30, +81), (31, –40), (31, +51), (32, –68), (32, +23), (33, –3), (33, +88), (34, –27), (34, +64), (36, –69), (36, +22), (37, –87), (37, +4), (38, –12), (38, +79), (40, –38), (40, +53), (41, –48), (41, +43), (43, –62), (43, +29), (44, –66), (44, +25), (45, –68), (45, +23), (46, –68), (46, +23), (47, –66), (47, +25), (48, –62), (48, +29), (50, –48), (50, +43), (51, –38), (51, +53), (53, –12), (53, +79), (54, –87), (54, +4), (55, –69), (55, +22), (57, –27), (57, +64), (58, –3), (58, +88), (59, –68), (59, +23), (60, –40), (60, +51), (61, –10), (61, +81), (62, –69), (62, +22), (64, –90), (64, +1), (66, –12), (66, +79), (67, –61), (67, +30), (68, –17), (68, +74), (69, –62), (69, +29), (71, –55), (71, +36), (72, –3), (72, +88), (73, –40), (73, +51), (74, –75), (74, +16), (75, –17), (75, +74), (76, –48), (76, +43), (79, –38), (79, +53), (80, –61), (80, +30), (81, –82), (81, +9), (82, –10), (82, +81), (83, –27), (83, +64), (85, –55), (85, +36), (86, –66), (86, +25), (87, –75), (87, +16), (88, –82), (88, +9), (89, –87), (89, +4), (90, –90), (90, +1)

95

(1, –94), (1, –63), (1, –47), (1, +1), (1, +32), (1, +48), (2, –93), (2, –91), (2, –62), (2, +2), (2, +4), (2, +33), (3, –92), (3, –86), (3, –43), (3, +3), (3, +9), (3, +52), (4, –87), (4, –79), (4, –58), (4, +8), (4, +16), (4, +37), (5, –93), (5, –17), (5, +2), (5, +78), (6, –83), (6, –77), (6, –59), (6, +12), (6, +18), (6, +36), (7, –47), (7, –46), (7, –23), (7, +48), (7, +49), (7, +72), (8, –63), (8, –42), (8, –31), (8, +32), (8, +53), (8, +64), (9, –68), (9, –14), (9, –7), (9, +27), (9, +81), (9, +88), (10, –87), (10, –68), (10, +8), (10, +27), (11, –82), (11, –69), (11, –23), (11, +13), (11, +26), (11, +72), (12, –47), (12, –46), (12, –23), (12, +48), (12, +49), (12, +72), (13, –58), (13, –21), (13, –7), (13, +37), (13, +74), (13, +88), (14, –93), (14, –92), (14, –89), (14, +2), (14, +3), (14, +6), (15, –77), (15, –58), (15, +18), (15, +37), (16, –73), (16, –62), (16, –29), (16, +22), (16, +33), (16, +66), (17, –93), (17, –91), (17, –62), (17, +2), (17, +4), (17, +33), (18, –82), (18, –56), (18, –28), (18, +13), (18, +39), (18, +67), (19, –93), (19, –83), (19, –73), (19, –68), (19, –63), (19, –58), (19, –43), (19, –28), (19, –23), (19, +2), (19, +12), (19, +22), (19, +27), (19, +32), (19, +37), (19, +52), (19, +67), (19, +72), (20, –82), (20, –63), (20, +13), (20, +32), (21, –43), (21, –34), (21, –17), (21, +52), (21, +61), (21, +78), (22, –92), (22, –86), (22, –43), (22, +3), (22, +9), (22, +52), (23, –77), (23, –68), (23, –41), (23, +18), (23, +27), (23, +54), (24, –93), (24, –92), (24, –89), (24, +2), (24, +3), (24, +6), (25, –83), (25, –7), (25, +12), (25, +88), (26, –84), (26, –42), (26, –28), (26, +11), (26, +53), (26, +67), (27, –63), (27, –42), (27, –31), (27, +32), (27, +53), (27, +64), (28, –87), (28, –83), (28, –71), (28, +8), (28, +12), (28, +24), (29, –68), (29, –14), (29, –7), (29, +27), (29, +81), (29, +88), (30, –42), (30, –23), (30, +53), (30, +72), (31, –84), (31, –42), (31, –28), (31, +11), (31, +53), (31, +67), (32, –58), (32, –21), (32, –7), (32, +37), (32, +74), (32, +88), (33, –73), (33, –51), (33, –17), (33, +22), (33, +44), (33, +78), (34, –87), (34, –79), (34, –58), (34, +8), (34, +16), (34, +37), (35, –92), (35, –73), (35, +3), (35, +22), (36, –43), (36, –34), (36, –17), (36, +52), (36, +61), (36, +78), (37, –82), (37, –56), (37, –28), (37, +13), (37, +39), (37, +67), (38, –92), (38, –87), (38, –82), (38, –77), (38, –62), (38, –47), (38, –42), (38, –17), (38, –7), (38, +3), (38, +8), (38, +13), (38, +18), (38, +33), (38, +48), (38, +53), (38, +78), (38, +88), (39, –94), (39, –63), (39, –47), (39, +1), (39, +32), (39, +48), (40, –62), (40, –43), (40, +33), (40, +52), (41, –73), (41, –62), (41, –29), (41, +22), (41, +33), (41, +66), (42, –77), (42, –68), (42, –41), (42, +18), (42, +27), (42, +54), (43, –73), (43, –51), (43, –17), (43, +22), (43, +44), (43, +78), (44, –83), (44, –77), (44, –59), (44, +12), (44, +18), (44, +36), (45, –47), (45, –28), (45, +48), (45, +67), (46, –82), (46, –69), (46, –23), (46, +13), (46, +26), (46, +72), (47, –87), (47, –83), (47, –71), (47, +8), (47, +12), (47, +24), (48, –87), (48, –83), (48, –71), (48, +8), (48, +12), (48, +24), (49, –82), (49, –69), (49, –23), (49, +13), (49, +26), (49, +72), (50, –47), (50, –28), (50, +48), (50, +67), (51, –83), (51, –77), (51, –59), (51, +12), (51, +18), (51, +36), (52, –73), (52, –51), (52, –17), (52, +22), (52, +44), (52, +78), (53, –77), (53, –68), (53, –41), (53, +18), (53, +27), (53, +54), (54, –73), (54, –62), (54, –29), (54, +22), (54, +33), (54, +66), (55, –62), (55, –43), (55, +33), (55, +52), (56, –94), (56, –63), (56, –47), (56, +1), (56, +32), (56, +48), (57, –92), (57, –87), (57, –82), (57, –77), (57, –62), (57, –47), (57, –42), (57, –17), (57, –7), (57, +3), (57, +8), (57, +13), (57, +18), (57, +33), (57, +48), (57, +53), (57, +78), (57, +88), (58, –82), (58, –56), (58, –28), (58, +13), (58, +39), (58, +67), (59, –43), (59, –34), (59, –17), (59, +52), (59, +61), (59, +78), (60, –92), (60, –73), (60, +3), (60, +22), (61, –87), (61, –79), (61, –58), (61, +8), (61, +16), (61, +37), (62, –73), (62, –51), (62, –17), (62, +22), (62, +44), (62, +78), (63, –58), (63, –21), (63, –7), (63, +37), (63, +74), (63, +88), (64, –84), (64, –42), (64, –28), (64, +11), (64, +53), (64, +67), (65, –42), (65, –23), (65, +53), (65, +72), (66, –68), (66, –14), (66, –7), (66, +27), (66, +81), (66, +88), (67, –87), (67, –83), (67, –71), (67, +8), (67, +12), (67, +24), (68, –63), (68, –42), (68, –31), (68, +32), (68, +53), (68, +64), (69, –84), (69, –42), (69, –28), (69, +11), (69, +53), (69, +67), (70, –83), (70, –7), (70, +12), (70, +88), (71, –93), (71, –92), (71, –89), (71, +2), (71, +3), (71, +6), (72, –77), (72, –68), (72, –41), (72, +18), (72, +27), (72, +54), (73, –92), (73, –86), (73, –43), (73, +3), (73, +9), (73, +52), (74, –43), (74, –34), (74, –17), (74, +52), (74, +61), (74, +78), (75, –82), (75, –63), (75, +13), (75, +32), (76, –93), (76, –83), (76, –73), (76, –68), (76, –63), (76, –58), (76, –43), (76, –28), (76, –23), (76, +2), (76, +12), (76, +22), (76, +27), (76, +32), (76, +37), (76, +52), (76, +67), (76, +72), (77, –82), (77, –56), (77, –28), (77, +13), (77, +39), (77, +67), (78, –93), (78, –91), (78, –62), (78, +2), (78, +4), (78, +33), (79, –73), (79, –62), (79, –29), (79, +22), (79, +33), (79, +66), (80, –77), (80, –58), (80, +18), (80, +37), (81, –93), (81, –92), (81, –89), (81, +2), (81, +3), (81, +6), (82, –58), (82, –21), (82, –7), (82, +37), (82, +74), (82, +88), (83, –47), (83, –46), (83, –23), (83, +48), (83, +49), (83, +72), (84, –82), (84, –69), (84, –23), (84, +13), (84, +26), (84, +72), (85, –87), (85, –68), (85, +8), (85, +27), (86, –68), (86, –14), (86, –7), (86, +27), (86, +81), (86, +88), (87, –63), (87, –42), (87, –31), (87, +32), (87, +53), (87, +64), (88, –47), (88, –46), (88, –23), (88, +48), (88, +49), (88, +72), (89, –83), (89, –77), (89, –59), (89, +12), (89, +18), (89, +36), (90, –93), (90, –17), (90, +2), (90, +78), (91, –87), (91, –79), (91, –58), (91, +8), (91, +16), (91, +37), (92, –92), (92, –86), (92, –43), (92, +3), (92, +9), (92, +52), (93, –93), (93, –91), (93, –62), (93, +2), (93, +4), (93, +33), (94, –94), (94, –63), (94, –47), (94, +1), (94, +32), (94, +48)

 

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