Don't take life too seriously: it's just a temporary situation

Nel 1980 Samuel Yates battezzò “primi unici” i numeri primi p, tali che la rappresentazione di 1 / p in base b abbia un periodo di una lunghezza che nessun altro primo ha.

Per esempio, in base 10 sono unici 3, 11, 37 e 101, perché 1 / 3, 1 / 111 / 371 / 101 hanno un periodo rispettivamente di 1, 2, 3 e 4 cifre e nessun altro primo ha un periodo della stessa lunghezza. Non è invece unico 7, perché sia 1 / 7, sia 1 / 13 hanno un periodo di 6 cifre.

 

In ogni base b, i primi con periodo di lunghezza n sono tra i divisori di Somma delle potenze k-esime di b, per k da 0 a n – 1, ossia del numero che in base b si rappresenta con n unità, e quindi sono in numero finito. Per esempio, in base 10 i primi con periodo di lunghezza 9 sono tra i divisori di 111111111 = 32 • 37 • 333667; tra questi, solo 333667 ha effettivamente periodo 9, quindi 333667 è unico ed è il solo primo con periodo di lunghezza 9 in base 10.

Dato che Somma delle potenze k-esime di b, per k da 0 a n – 1 uguale a (b^n – 1) / (b – 1) e che, con la sola eccezione di 26 – 1, per ogni valore di n > 2 vi è un fattore primo primitivo, ossia un primo che divide bn – 1, ma nessun numero della forma bk – 1 con k < n (teorema di Zsigmondy, v. potenze), per ogni base vi è almeno un primo con un periodo di qualsiasi lunghezza n > 2, salvo che non esistono primi con periodo di lunghezza 6 in base 2.

I primi con periodo di lunghezza 2 esistono in tutte le basi che non siano della forma ak – 1, con k > 1.

I fattori di b – 1 hanno periodo di lunghezza 1, quindi esistono, in numero finito, in qualsiasi base tranne 2.

Riassumendo, esiste un fattore primo con periodo di lunghezza n per ogni valore di ne ogni base, tranne che:

  • in base 2 non esistono primi con periodo di lunghezza 1 e 6;

  • in basi della forma ak – 1, con k > 1 non esistono primi con periodo di lunghezza 2.

 

Non è però detto che i fattori primi primitivi siano unici; un primo p è unico di periodo n in base b se e solo se Φn(b) / MCD(Φn(b), n) è una potenza di p, dove Φn(x) è l’n-esimo polinomio ciclotomico. Di conseguenza determinare l’esistenza di un primo unico di lunghezza n in base b è teoricamente semplice: si prende (b^n – 1) / (b – 1) e lo si divide per tutti i fattori primi dei numeri della forma (b^m – 1) / (b – 1) (rappresentati con m volte 1 nella stessa base), per ogni divisore m di n minore di n; il quoziente è il prodotto dei primi con periodo n e se è una potenza di un primo (eventualmente con esponente 1), il primo è unico, con periodo di lunghezza n.

Per esempio, nel caso n = 12 in base 10, il numero formato da 12 volte 1 è 111111111111 = 3 • 7 • 11 • 13 • 37 • 101 • 9901; i divisori di 12 sono 2, 3, 4 e 6 e i corrispondenti numeri formati da sequenze di 1 sono 11 = 11, 111 = 3 • 37, 1111 = 11 • 101 e 111111 = 3 • 7 • 11 • 13 • 37. Eliminando i divisori di questi numeri dalla scomposizione di 111111111111 resta il solo 9901, che quindi è un primo unico in base 10, con periodo di lunghezza 12.

Il metodo però richiede la scomposizione in fattori primi di un numero di n cifre e l’operazione può essere molto complessa per n anche solo dell’ordine delle centinaia.

In base 10 l’unico caso noto in cui il quoziente sia una potenza di un primo con esponente maggiore di 1 si ha con n = 1: Φ1(10) / MCD(Φ1(10), 1) = 9; molto probabilmente non ce ne sono altri.

 

In qualsiasi base tutti i primi pluriunitari sono primi unici e a lungo periodo.

 

La tabella seguente riporta le minime lunghezze dei periodi per i quali esistano primi unici, nelle basi fino a 24.

Base

Lunghezze

2

2, 3, 4, 5, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 24, 26, 27, 30, 31, 32, 33, 34, 38, 40, 42, 46, 49, 54, 56, 61, 62, 65, 69, 77, 78, 80, 85, 86, 89, 90, 93, 98, 107, 120, 122, 126, 127, 129, 133, 145, 147, 150, 158, 165, 170, 174, 184, 192, 195, 202, 208, 234, 254, 261, 280, 296, 312, 322, 334, 342, 345, 366, 374, 382, 398, 410, 414, 425, 447, 471, 507, 521, 550, 567

2

2, 3, 4, 5, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 24, 26, 27, 30, 31, 32, 33, 34, 38, 40, 42, 46, 49, 54, 56, 61, 62, 65, 69, 77, 78, 80, 85, 86, 89, 90, 93, 98, 107, 120, 122, 126, 127, 129, 133, 145, 147, 150, 158, 165, 170, 174, 184, 192, 195, 202, 208, 234, 254, 261, 280, 296, 312, 322, 334, 342, 345, 366, 374, 382, 398, 410, 414, 425, 447, 471, 507, 521, 550, 567

3

1, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 20, 21, 24, 26, 32, 33, 36, 40, 46, 60, 63, 64, 70, 71, 72, 86, 103, 108, 128, 130, 132, 143, 145, 154, 161, 236, 255, 261, 276, 279, 287, 304, 364, 430, 464, 513, 528, 541, 562

4

1, 2, 3, 4, 6, 8, 10, 12, 16, 20, 28, 40, 60, 92, 96, 104, 140, 148, 156, 300, 356, 408

5

1, 2, 3, 4, 6, 7, 8, 10, 11, 12, 13, 18, 24, 28, 47, 48, 49, 56, 57, 88, 90, 92, 108, 110, 116, 120, 127, 134, 141, 149, 161, 171, 181, 198, 202, 206, 236, 248, 288, 357, 384, 420, 458, 500, 530, 536

6

1, 2, 3, 4, 5, 6, 7, 8, 18, 21, 22, 24, 29, 30, 42, 50, 62, 71, 86, 90, 94, 118, 124, 127, 129, 144, 154, 186, 192, 214, 271, 354, 360, 411, 480, 509, 558, 575

7

3, 4, 5, 6, 8, 13, 18, 21, 28, 30, 34, 36, 46, 48, 50, 54, 55, 58, 63, 76, 84, 94, 105, 122, 131, 148, 149, 224, 280, 288, 296, 332, 352, 456, 528, 531

8

1, 2, 3, 6, 9, 18, 30, 42, 78, 87, 114, 138, 189, 303, 318, 330, 408, 462, 504, 561

9

1, 2, 4, 6, 10, 12, 16, 18, 20, 30, 32, 36, 54, 64, 66, 118, 138, 152, 182, 232, 264, 336, 340, 380, 414, 446, 492, 540

10

1, 2, 3, 4, 9, 10, 12, 14, 19, 23, 24, 36, 38, 39, 48, 62, 93, 106, 120, 134, 150, 196, 294, 317, 320, 385

11

2, 4, 5, 6, 8, 9, 10, 14, 15, 17, 18, 19, 20, 27, 36, 42, 45, 52, 60, 73, 91, 104, 139, 205, 234, 246, 318, 358, 388, 403, 458, 552

12

1, 2, 3, 5, 10, 12, 19, 20, 21, 22, 56, 60, 63, 70, 80, 84, 92, 97, 109, 111, 123, 164, 189, 218, 276, 317, 353, 364, 386, 405, 456, 511

13

2, 3, 5, 6, 7, 8, 9, 12, 16, 22, 24, 28, 33, 34, 38, 78, 80, 102, 137, 140, 147, 224, 230, 283, 304, 341, 360, 372, 384, 418, 420, 436, 483, 568, 570

14

1, 3, 4, 6, 7, 14, 19, 24, 31, 33, 35, 36, 41, 55, 60, 106, 114, 129, 152, 153, 172, 222, 265, 286, 400, 448, 560

15

3, 4, 6, 7, 14, 24, 43, 54, 58, 73, 85, 93, 102, 184, 220, 221, 228, 232, 247, 291, 305, 486, 487, 505, 551, 552

16

2, 4, 6, 8, 10, 14, 20, 30, 46, 48, 52, 70, 74, 78, 150, 178, 204, 298, 306, 346, 366, 378, 400, 476, 498, 502

17

1, 2, 3, 5, 7, 8, 11, 12, 14, 15, 34, 42, 46, 47, 48, 50, 71, 77, 94, 110, 114, 147, 154, 176, 228, 235, 258, 275, 338, 350, 419, 450, 480, 515

18

1, 2, 3, 6, 14, 17, 21, 24, 30, 33, 38, 45, 46, 72, 78, 114, 146, 168, 288, 414, 440, 448

19

2, 3, 4, 6, 19, 20, 31, 34, 47, 56, 59, 61, 70, 74, 91, 92, 96, 98, 107, 120, 145, 156, 168, 242, 276, 314, 326, 337, 387, 565

20

1, 3, 4, 6, 8, 9, 10, 11, 17, 30, 98, 100, 110, 126, 154, 158, 160, 168, 178, 182, 228, 266, 270, 280, 340, 416, 480, 574

21

2, 3, 5, 6, 8, 9, 10, 11, 14, 17, 26, 43, 64, 74, 81, 104, 192, 271, 321, 335, 348, 404, 437, 445, 516

22

2, 5, 6, 7, 10, 21, 25, 26, 69, 79, 86, 93, 100, 101, 154, 158, 161, 171, 202, 214, 294, 354, 359, 424, 454

23

2, 5, 8, 11, 15, 22, 26, 39, 42, 45, 54, 56, 132, 134, 145, 147, 196, 212, 218, 252, 343

24

1, 2, 3, 4, 5, 8, 14, 19, 22, 38, 45, 53, 54, 70, 71, 117, 140, 144, 169, 186, 192, 195, 196, 430

 

I primi unici sono estremamente rari: in base 10 vi sono oltre 1047 primi minori di 1050, ma solo 18 sono unici. Si ritiene però che siano infiniti in qualsiasi base.

 

Il massimo primo unico noto in base 10 è Φ23749(10) / MCD(Φ23749(10), 23749) (Chandler, 2014), ha 20160 cifre; il massimo primo unico probabile è il massimo primo pluriunitario probabile noto, formato da 270343 volte 1.

Il massimo primo unico noto (in base 2) è il massimo primo di Mersenne noto, ossia 274207281 – 1.

Il massimo primo unico noto in una base diversa da 2 è Φ123447^524288(3) / MCD(Φ123447^524288(3), 123447^524288), in base 3, (Propper e Batalov, 2017), ha 5338805 cifre.

 

La tabella mostra i minimi primi unici in base 10.

Primo

Periodo

3

1

11

2

37

3

101

4

9091

10

9901

12

333667

9

909091

14

99990001

24

999999000001

36

9999999900000001

48

909090909090909091

38

1111111111111111111

19

11111111111111111111111

23

900900900900990990990991

39

909090909090909090909090909091

62

100009999999899989999000000010001

120

10000099999999989999899999000000000100001

150

9090909090909090909090909090909090909090909090909091

106

900900900900900900900900900900990990990990990990990990990991

93

909090909090909090909090909090909090909090909090909090909090909091

134

142857157142857142856999999985714285714285857142857142855714285571428571428572857143

294

999999999999990000000000000099999999999999000000000000009999999999999900000000000001

196

99999999999999999999999999999999000000000000000000000000000000009999999999999999999999999999999900000000000000000000000000000001

320

1098901098901098901098901098901098901098901098901098901098901098901098901098901098901098901098901098901098900989010989010989010989010989010989010989010989010989010989010989010989010989010989010989010989010989010989011

654

1000999998999000001000999998999000001000999998999000001000999998999000001000999998999000001000999998999000001000999998998999000001000999998999000001000999998999000001000999998999000001000999998999000001000999998999000001000999998999000001001

738

1111099888878889001111100000000000011110998888788890011222109988887889011221109998887788901111110999878890112222210987766677901122222109888790001111110988877889000111221098887888900112221099888878889001111100000000000011110998888788890011111

385

9090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909091

586

11111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111

317

900900900900900900900900900900900900900900900900900900900900900900900900900900900900900900900900900900900900900900900900900900900900900900900900900900900900900900900900900900900900900900900900900900990990990990990990990990990990990990990990990990990990990990990990990990990990990990990990990990990990990990990990990990990990990990990990990990990990990990990990990990990990990990990990990990990991

597

1000000000000000000999999999999999999999999999999999998999999999999999999000000000000000000000000000000000001000000000000000000999999999999999999999999999999999998999999999999999999000000000000000000000000000000000000999999999999999999999999999999999998999999999999999999000000000000000000000000000000000001000000000000000000999999999999999999999999999999999998999999999999999999000000000000000000000000000000000001000000000000000001

1404

1000000001000000000999999999999999999999999998999999998999999997999999998999999999000000000000000000000000001000000001000000001000000001000000001000000000999999999999999999999999998999999999999999998999999999999999998999999999999999998999999999999999999000000000000000000000000001000000001000000001000000001000000001000000000999999999999999999999999998999999998999999997999999998999999999000000000000000000000000001000000001000000001

945

100000000000000000000009999999999999999999999999999999999999999999899999999999999999999990000000000000000000000000000000000000000000100000000000000000000009999999999999999999999999999999999999999999899999999999999999999989999999999999999999999000000000000000000000000000000000000000000010000000000000000000000999999999999999999999999999999999999999999989999999999999999999999000000000000000000000000000000000000000000010000000000000000000001

1452

1000000000000000000999999999999999999999999999999999998999999999999999999000000000000000000000000000000000001000000000000000000999999999999999999999999999999999998999999999999999999000000000000000000000000000000000001000000000000000000999999999999999999999999999999999998999999999999999998999999999999999999000000000000000000000000000000000001000000000000000000999999999999999999999999999999999998999999999999999999000000000000000000000000000000000001000000000000000000999999999999999999999999999999999998999999999999999999000000000000000000000000000000000001000000000000000001

1836

1000099999998999900000001000099999998999900000001000099999998999900000001000099999998999900000001000099999998999900000001000099999998999900000001000099999998999900000001000099999998999900000001000099999998999900000001000099999998999900000001000099999998999900000001000099999998999900000000999999989999000000010000999999989999000000010000999999989999000000010000999999989999000000010000999999989999000000010000999999989999000000010000999999989999000000010000999999989999000000010000999999989999000000010000999999989999000000010000999999989999000000010000999999989999000000010001

1752

99009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009901

1172

1009998990001009998990001009998990001009998990001009998990001009998990001009998990001009998990001009998990001009998990001009998990001009998990001009998990001009998990001009998990001009998990001009998990001009998990001009998990001009998990001009998990001009998990001009998990001009998990001009998990000999899000100999899000100999899000100999899000100999899000100999899000100999899000100999899000100999899000100999899000100999899000100999899000100999899000100999899000100999899000100999899000100999899000100999899000100999899000100999899000100999899000100999899000100999899000100999899000100999899000101

1812

9090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909091

1282

1099999999999999999999988999999890000000000000110000001100000010999998899999988999999890000009900000110000001099999900999999009999989000000990000009900000098999990099999900999999010000100100000990000009899998998999989989999901000010010000100100001000999899899998998999989990000999900010010000100099990000999900009998999000099990000999900009998999000099990000999900010010000100099990000999899899998998999989990001001000010010000100100000989999899899998998999990099999901000010010000099000000990000009899999009999990099999900999998900000099000000990000011000000109999990099999889999998899999989000001100000011000000109999999999999889999998900000000000000000000011

1426

1000000000000999999999999999999999998999999999999000000000000000000000001000000000000999999999999999999999998999999999999000000000000000000000001000000000000999999999999999999999998999999999999000000000000000000000001000000000000999999999999999999999998999999999999000000000000000000000001000000000000999999999999999999999998999999999999000000000000000000000000999999999999999999999998999999999999000000000000000000000001000000000000999999999999999999999998999999999999000000000000000000000001000000000000999999999999999999999998999999999999000000000000000000000001000000000000999999999999999999999998999999999999000000000000000000000001000000000000999999999999999999999998999999999999000000000000000000000001000000000001

2232

1000000099999999999999999999999999999999999999999899999990000000000000000000000000000000000000000010000000999999999999999999999999999899999989999998999999900000000000000000000000000010000001000000100000009999999999999999999999999998999999899999989999999000000000000010000001000000100000010000001000000099999999999998999999899999989999998999999899999990000000000000100000010000001000000100000009999999999999899999989999998999999899999989999999000000000000010000001000000100000010000001000000099999999999998999999899999989999999000000000000000000000000000100000010000001000000099999999999999999999999999989999998999999899999990000000000000000000000000001000000099999999999999999999999999999999999999999899999990000000000000000000000000000000000000000010000001

1862

99009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009901

1844

999999999999999999999999999999999999999000000000000000000000000000000000000000000000000000000000000000000000000000000999999999999999999999999999999999999999000000000000000000000000000000000000000000000000000000000000000000000000000000999999999999999999999999999999999999999000000000000000000000000000000000000000000000000000000000000000000000000000000999999999999999999999999999999999999999000000000000000000000000000000000000000000000000000000000000000000000000000000999999999999999999999999999999999999999999999999999999999999999999999999999999000000000000000000000000000000000000000999999999999999999999999999999999999999999999999999999999999999999999999999999000000000000000000000000000000000000000999999999999999999999999999999999999999999999999999999999999999999999999999999000000000000000000000000000000000000000999999999999999999999999999999999999999999999999999999999999999999999999999999000000000000000000000000000000000000001

1521

1099999999989000000000109999999998900000000010999999999890000000001099999999989000000000109999999888900000001110999999988890000000111099999998889000000011109999999888900000001110999999988890000011111099999888889000001111109999988888900000111110999998888890000011111099999888889000001111109998888888900011111110999888888890001111111099988888889000111111109998888888900011111110999888888890111111111098888888889011111111109888888888901111111110988888888890111111111098888888889011111111098888888889011111111109888888888901111111110988888888890111111111098888888889011111111109888888890001111111099988888889000111111109998888888900011111110999888888890001111111099988888889000111110999998888890000011111099999888889000001111109999988888900000111110999998888890000011111099999888900000001110999999988890000000111099999998889000000011109999999888900000001110999999988890000000109999999998900000000010999999999890000000001099999999989000000000109999999998900000000011

2134

11111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111

3750

999900009999000099990000999900009999000099990000999900009999000099990000999900009999000099990000999900009999000099990000999900009999000099990000999900009999000099990000999900009999000099990000999900009999000099990000999900009999000099990000999900009999000099990000999900009999000099990000999900009999000099990000999900009999000099990000999900009999000099990000999900009999000099990000999900009999000099990000999900009999000099990000999900009999000099990000999900009999000099990000999900009999000099990000999900009999000099990000999900009999000099990000999900009999000099990000999900009999000099990000999900009999000099990000999900009999000099990000999900009999000099990000999900009999000099990000999900009999000099990000999900009999000099990000999900009999000099990000999900009999000099990000999900009999000099990000999900009999000099990000999900009999000099990000999900009999000099990000999900009999000099990000999900009999000099990000999900009999000099990000999900009999000099990000999900009999000099990000999900009999000099990000999900009999000099990000999900009999000099990000999900009999000099990000999900009999000099990001

1031

 

In base 2 sono primi unici, tra gli altri:

  • i primi di Fermat 22n + 1, con periodo di lunghezza 2n + 1;

  • i primi di Mersenne Mp (che sono pluriunitari in base 2), con periodo di lunghezza p;

  • i primi di Wagstaff Wp, con periodo di lunghezza 2p.

Non esistono invece primi unici in base 2 con periodo di lunghezza maggiore di 20 e della forma 8k + 4, perché esistono almeno due primi con periodo di tali lunghezze.

 

I massimi primi unici in base 2 sono i primi di Mersenne, seguiti dai primi di Wagstaff.

 

I primi unici in base 2 non dividono alcun overpseudoprimo in base 2.

 

Gli unici casi noti nei quali Φn(2) sia compostoΦn(2) / MCD(Φn(2), n) sia primo si hanno per n uguale a 18, 20, 21, 54, 147, 342, 602 e 889, corrispondenti rispettivamente ai primi 19, 41, 337, 87211, 2741672362528725535068727, 19177458387940268116349766612211, 250496677636134194455624482113419891241717626649461375803326671768162580233 e 1504004909926131633188840257128563607541163140104723054723183378190537555932072058265677602337213984792802468007992843498623739068694344880627731976582462714986041644019253711037305513830373917224858668705029882514901678735617; si suppone che non ne esistano altri.

Tutti gli altri primi unici noti in base 2 sono della forma Φn(2).

Non si conoscono casi nei quali Φn(2) / MCD(Φn(2), n) sia una potenza di un primo e si suppone non esistano.

 

La tabella mostra i primi unici minori di 264 in base 2.

Primo

Periodo

3

2

5

4

7

3

11

10

13

12

17

8

19

18

31

5

41

20

43

14

73

9

127

7

151

15

241

24

257

16

331

30

337

21

683

22

2731

26

5419

42

8191

13

43691

34

61681

40

65537

32

87211

54

131071

17

174763

38

262657

27

524287

19

599479

33

2796203

46

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90

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78

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62

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31

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80

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120

77158673929

126

1133836730401

150

2932031007403

86

4363953127297

98

4432676798593

49

10052678938039

69

145295143558111

65

96076791871613611

174

581283643249112959

77

658812288653553079

93

768614336404564651

122

2305843009213693951

61

9520972806333758431

85

18446744069414584321

192

 

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