¿Cuándo$2^{2m}+2^m+1$ divide$10^{2m}+10^m+1$?

Question

¿Para qué valores de $m\in \Bbb N$ es verdad que divide a que $2^{2m}+2^m+1$ $10^{2m}+10^m+1$?

Tengo falso $m=1,3 $ y verdadero $m=2,4$

Esto surge de mis pensamientos respecto a la Cómo encontrar todos los números binarios en base $10$ s.t. que su Divisibilidad por su propio valor numérico en base $2$?

Answer 1

respuesta parcial: con mi comentario que:
$\begin{align} \\ 2^{2m}+2^m+1 &\mid 2^{2m}5^{2m}+2^m5^m+1 \\ &\implies 2^{2m}+2^m+1 \mid 2^{2m}(5^{2m}-1)+2^m(5^m-1) \\ &\implies 2^{2m}+2^m+1 \mid 2^m\big({5^{2m}-1\over 4}\big)+\big({5^m-1\over 4}\big) \\ &\implies 2^m(2^m+1)+1 \mid 2^m\big({5^{2m}-1\over 4}\big)+\big({5^m-1\over 4}\big) \end {Alinee el} $

Espero que esto ayuda a simplificar las cosas. Edit: las fracciones son los 5 repunits base.

Answer 2

Resultado parcial que cubre aproximadamente dos tercios de todos los números de $m$.

Estoy buscando una que cubre el conjunto de los números primos. La idea es que cada vez que nos encontramos con un primer $p$ tal que $p\mid 4^m+2^m+1$ pero $p\nmid 100^m+10^m+1$ hemos demostrado que $4^m+2^m+1\nmid 100^m+10^m+1$. Cuando la caza de adecuado de los números primos $p$ podemos simplemente mirar el factorizations de estos números. Para $p>3$ a ser un factor de $k^{2m}+k^m+1=(k^{3m}-1)/(k^m-1)$ vemos que $k^m$ tiene que ser una primitiva de la tercera raíz de la unidad del modulo $p$. Obviamente aplicar esto a ambos $k=2$$k=10$.

Para conseguir esos primitivos tercer raíces de la unidad, necesitamos $\ell=\operatorname{ord}_p(2)$ ser divisible por tres. Si $\ell=3u$, vemos que $p\mid 4^m+2^m+1$ si y sólo si $m\equiv u,2u\pmod{3u}$. Pero, si $t=\operatorname{ord}_p(10)$ también es divisible por tres, decir $t=3v$, también tenemos que $p\mid 100^m+10^m+1$ siempre $m\equiv v,2v\pmod{3v}$. De todos modos, esto permite concluir que el $m$ no puede ser en ciertas clases de residuos modulo $3w$ donde $w=\operatorname{lcm}(u,v)$. Observar que si $t$ no es un múltiplo de tres, $100^m+10^m+1$ nunca es divisible por $p$, debido a $10^m$ es nunca una primitiva tercera raíz de la unidad.

Llegar nuestras manos sucias vemos que:

• Con $p=7$ tenemos $\ell=3$$t=6$, tanto divisible por tres. Esto significa que cuando se $m\equiv1,5\pmod6$ tenemos $7\mid 4^m+2^m+1$$7\nmid 100^m+10^m+1$.
• Con $p=73\mid 2^9-1$ tenemos $\ell=9=3\cdot3$$t=8$. Esto significa que $73\mid 4^m+2^m+1$ fib $m\equiv3,6\pmod9$. Pero $100^m+10^m+1$ nunca es divisible por $73$, por lo que hemos cubierto estos valores de $m$. En este punto, hemos excluido $10$ $18$ residuo de clases modulo $18$, por lo que hemos cubierto más de la mitad de los números enteros.
• El primer $p=13$ es un factor de $2^{12}-1$$\ell=12$. Pero aquí $t=6$ es un factor de $\ell$. Esta es una mala noticia para nosotros. No hay valores de $m$ puede ser excluido. Por si $m\equiv4,8\pmod{12}$ automáticamente $m\equiv2,4\pmod{6}$. En estos casos $13$ es un factor común de ambos $4^m+2^m+1$$100^m+10^m+1$. Lo mismo sucede cada vez que $2$ es una raíz primitiva módulo $p$. Realmente necesitamos buscar no-primitiva de los casos.
• Con $p=151\mid 2^{15}-1$ tenemos $\ell=15=3\cdot5$. Aquí $t=75=3\cdot25$. Esto significa que podemos descartar $m$ si $m\equiv5,10\pmod{15}$$m\not\equiv25,50\pmod{75}$.
• Con $p=241\mid 2^{24}-1$ tenemos $\ell=24=3\cdot8$. Aquí $t=30=3\cdot10$. Podemos regla ouf $m$ si $m\equiv8,16\pmod{24}$$m\not\equiv10,20\pmod{30}$.

En este punto hemos descartado $1216$ residuo de clases modulo $1800$, por lo que hemos llegado a los dos tercios de punto. Que el módulo de $1800$ es el mínimo común múltiplo de a $6,9,75,24$ $30$ que aparece arriba. Recordando que $m=0,2,4$ son soluciones de la más pequeña de los valores de $m$ que aún no hemos cubierto se $m=9,14,18,22$. Parar aquí por ahora.

Answer 3

hasta 100, el extraño $m$ para que el número 2 y la número 10 no son coprimos son $m=25,33,67,$ mientras estos gcd es aún muy pequeñas en comparación con los números crudos.

1
111 = 3  37
7 =  7
1  GCD 1  m  is ODD  

2
10101 = 3 7 13  37
21 = 3  7
2  GCD 21
  DIVIDES  

3
1001001 = 3  333667
73 =  73
3  GCD 1  m  is ODD  

4
100010001 = 3 7 13 37  9901
273 = 3 7  13
4  GCD 273
  DIVIDES  

5
10000100001 = 3 31 37  2906161
1057 = 7  151
5  GCD 1  m  is ODD  

6
1000001000001 = 3 19 52579  333667
4161 = 3 19  73
6  GCD 57

7
100000010000001 = 3 37 43 1933  10838689
16513 = 7^2  337
7  GCD 1  m  is ODD  

8
10000000100000001 = 3 7 13 37 9901  99990001
65793 = 3 7 13  241
8  GCD 273

9
1000000001000000001 = 3 757  cdot mbox{BIG} 
262657 =  262657
9  GCD 1  m  is ODD  

10
100000000010000000001 = 3 7 13 31 37 211 241 2161  2906161
1049601 = 3 7 151  331
10  GCD 21

11
10000000000100000000001 = 3 37 67  cdot mbox{BIG} 
4196353 = 7  599479
11  GCD 1  m  is ODD  

12
1000000000001000000000001 = 3 19 52579 333667  999999000001
16781313 = 3 19 37 73  109
12  GCD 57

13
100000000000010000000000001 = 3 37  cdot mbox{BIG} 
67117057 = 7 79  121369
13  GCD 1  m  is ODD  

14
10000000000000100000000000001 = 3 7^2 13 37 43 127 1933 2689 459691  10838689
268451841 = 3 7^2 337  5419
14  GCD 147

15
1000000000000001000000000000001 = 3 238681 333667  cdot mbox{BIG} 
1073774593 = 73 631  23311
15  GCD 1  m  is ODD  

16
100000000000000010000000000000001 = 3 7 13 37 9901  cdot mbox{BIG} 
4295032833 = 3 7 13 97 241  673
16  GCD 273

17
10000000000000000100000000000000001 = 3 37 613 210631  cdot mbox{BIG} 
17180000257 = 7 103 2143  11119
17  GCD 1  m  is ODD  

18
1000000000000000001000000000000000001 = 3 757  cdot mbox{BIG} 
68719738881 = 3 87211  262657
18  GCD 3

19
100000000000000000010000000000000000001 = 3 37 21319  cdot mbox{BIG} 
274878431233 = 7 32377  1212847
19  GCD 1  m  is ODD  

20
10000000000000000000100000000000000000001 = 3 7 13 31 37 61 211 241 2161 9901  cdot mbox{BIG} 
1099512676353 = 3 7 13 61 151 331  1321
20  GCD 16653

21
1000000000000000000001000000000000000000001 = 3 10837 23311 45613 333667  cdot mbox{BIG} 
4398048608257 = 73 92737  649657
21  GCD 1  m  is ODD  

22
100000000000000000000010000000000000000000001 = 3 7 13 37 67  cdot mbox{BIG} 
17592190238721 = 3 7 67 20857  599479
22  GCD 1407

23
10000000000000000000000100000000000000000000001 = 3 37 277  cdot mbox{BIG} 
70368752566273 = 7  cdot mbox{BIG} 
23  GCD 1  m  is ODD  

24
1000000000000000000000001000000000000000000000001 = 3 19 3169 52579 98641 333667  cdot mbox{BIG} 
281474993487873 = 3 19 37 73 109 433  38737
24  GCD 57

25
100000000000000000000000010000000000000000000000001 = 3 31 37 151 4201  cdot mbox{BIG} 
1125899940397057 = 7 151 100801  10567201
25  GCD 151  m  is ODD  

26
10000000000000000000000000100000000000000000000000001 = 3 7 13^2 37 157 6397 216451  cdot mbox{BIG} 
4503599694479361 = 3 7 79 121369  22366891
26  GCD 21

27
1000000000000000000000000001000000000000000000000000001 = 3 163 9397  cdot mbox{BIG} 
18014398643699713 = 2593 71119  97685839
27  GCD 1  m  is ODD  

28
100000000000000000000000000010000000000000000000000000001 = 3 7^2 13 37 43 127 1933 2689 9901 226549 459691  cdot mbox{BIG} 
72057594306363393 = 3 7^2 13 337 1429 5419  14449
28  GCD 1911

29
10000000000000000000000000000100000000000000000000000000001 = 3 37 4003 72559  cdot mbox{BIG} 
288230376688582657 = 7 4177  cdot mbox{BIG} 
29  GCD 1  m  is ODD  

30
1000000000000000000000000000001000000000000000000000000000001 = 3 19 29611 52579 238681 333667  cdot mbox{BIG} 
1152921505680588801 = 3 19 73 631 23311  18837001
30  GCD 57

31
100000000000000000000000000000010000000000000000000000000000001 = 3 37  cdot mbox{BIG} 
4611686020574871553 = 7  cdot mbox{BIG} 
31  GCD 1  m  is ODD  

32
10000000000000000000000000000000100000000000000000000000000000001 = 3 7 13 37 97 9901 206209  cdot mbox{BIG} 
18446744078004518913 = 3 7 13 97 193 241 673  22253377
32  GCD 26481

33
1000000000000000000000000000000001000000000000000000000000000000001 = 3 199 397 34849 333667  cdot mbox{BIG} 
73786976303428141057 = 73 199 153649  33057806959
33  GCD 199  m  is ODD  

34
100000000000000000000000000000000010000000000000000000000000000000001 = 3 7 13 37 613 210631  cdot mbox{BIG} 
295147905196532695041 = 3 7 103 307 2143 2857 6529  11119
34  GCD 21

35
10000000000000000000000000000000000100000000000000000000000000000000001 = 3 31 37 43 1933  cdot mbox{BIG} 
1180591620751771041793 = 7^2 151 337 29191 106681  152041
35  GCD 1  m  is ODD  

36
1000000000000000000000000000000000001000000000000000000000000000000000001 = 3 109 757 153469  cdot mbox{BIG} 
4722366482938364690433 = 3 87211 246241 262657  279073
36  GCD 3

37
100000000000000000000000000000000000010000000000000000000000000000000000001 = 3 37^2  cdot mbox{BIG} 
18889465931616019808257 = 7 321679  cdot mbox{BIG} 
37  GCD 1  m  is ODD  

38
10000000000000000000000000000000000000100000000000000000000000000000000000001 = 3 7 13 37 21319  cdot mbox{BIG} 
75557863726189201326081 = 3 7 571 32377  cdot mbox{BIG} 
38  GCD 21

39
1000000000000000000000000000000000000001000000000000000000000000000000000000001 = 3 333667  cdot mbox{BIG} 
302231454904207049490433 = 73 937 6553 86113  7830118297
39  GCD 1  m  is ODD  

40
100000000000000000000000000000000000000010000000000000000000000000000000000000001 = 3 7 13 31 37 61 211 241 2161 9901  cdot mbox{BIG} 
1208925819615728686333953 = 3 7 13 61 151 241 331 1321  4562284561
40  GCD 4013373

41
10000000000000000000000000000000000000000100000000000000000000000000000000000000001 = 3 37  cdot mbox{BIG} 
4835703278460715722080257 = 7  cdot mbox{BIG} 
41  GCD 1  m  is ODD  

42
1000000000000000000000000000000000000000001000000000000000000000000000000000000000001 = 3 19 10837 23311 45613 52579 333667  cdot mbox{BIG} 
19342813113838464841809921 = 3 19 73 92737 649657  77158673929
42  GCD 57

43
100000000000000000000000000000000000000000010000000000000000000000000000000000000000001 = 3 37  cdot mbox{BIG} 
77371252455345063274217473 = 7  cdot mbox{BIG} 
43  GCD 1  m  is ODD  

44
10000000000000000000000000000000000000000000100000000000000000000000000000000000000000001 = 3 7 13 37 67 9901  cdot mbox{BIG} 
309485009821362660910825473 = 3 7 13 67 20857 312709 599479  4327489
44  GCD 18291

45
1000000000000000000000000000000000000000000001000000000000000000000000000000000000000000001 = 3 757  cdot mbox{BIG} 
1237940039285415459271213057 = 271 262657 348031  cdot mbox{BIG} 
45  GCD 1  m  is ODD  

46
100000000000000000000000000000000000000000000010000000000000000000000000000000000000000000001 = 3 7 13 37 277 31051  cdot mbox{BIG} 
4951760157141591468340674561 = 3 7 139  cdot mbox{BIG} 
46  GCD 21

47
10000000000000000000000000000000000000000000000100000000000000000000000000000000000000000000001 = 3 37 283  cdot mbox{BIG} 
19807040628566225135874342913 = 7  cdot mbox{BIG} 
47  GCD 1  m  is ODD  

48
1000000000000000000000000000000000000000000000001000000000000000000000000000000000000000000000001 = 3 19 3169 8929 52579 98641 333667  cdot mbox{BIG} 
79228162514264619068520660993 = 3 19 37 73 109 433 577 38737  487824887233
48  GCD 57

49
100000000000000000000000000000000000000000000000010000000000000000000000000000000000000000000000001 = 3 37 43 1933 63799  cdot mbox{BIG} 
316912650057057913324129222657 = 7^3 337  cdot mbox{BIG} 
49  GCD 1  m  is ODD  

50
10000000000000000000000000000000000000000000000000100000000000000000000000000000000000000000000000001 = 3 7 13 31 37 151 211 241 2161 4201  cdot mbox{BIG} 
1267650600228230527396610048001 = 3 7 151 331 100801  cdot mbox{BIG} 
50  GCD 3171

51
1000000000000000000000000000000000000000000000000001000000000000000000000000000000000000000000000000001 = 3 307 18973 333667  cdot mbox{BIG} 
5070602400912919857786626506753 = 73 919  cdot mbox{BIG} 
51  GCD 1  m  is ODD  

52
100000000000000000000000000000000000000000000000000010000000000000000000000000000000000000000000000000001 = 3 7 13^2 37 157 3121 6397 9901 216451  cdot mbox{BIG} 
20282409603651674927546878656513 = 3 7 13^2 79 313 1249 3121 21841 121369  22366891
52  GCD 11076429

53
10000000000000000000000000000000000000000000000000000100000000000000000000000000000000000000000000000000001 = 3 37 351391  cdot mbox{BIG} 
81129638414606690702988259885057 = 7 6679  cdot mbox{BIG} 
53  GCD 1  m  is ODD  

54
1000000000000000000000000000000000000000000000000000001000000000000000000000000000000000000000000000000000001 = 3 163 1459 9397  cdot mbox{BIG} 
324518553658426744797554530058241 = 3 163 2593 71119 135433  cdot mbox{BIG} 
54  GCD 489

55
100000000000000000000000000000000000000000000000000000010000000000000000000000000000000000000000000000000000001 = 3 31 37 67 471241  cdot mbox{BIG} 
1298074214633706943161421101268993 = 7 151 599479  cdot mbox{BIG} 
55  GCD 1  m  is ODD  

56
10000000000000000000000000000000000000000000000000000000100000000000000000000000000000000000000000000000000000001 = 3 7^2 13 37 43 127 1933 2689 9901 226549 459691  cdot mbox{BIG} 
5192296858534827700588090367148033 = 3 7^2 13 241 337 1429 3361 5419 14449  88959882481
56  GCD 1911

57
1000000000000000000000000000000000000000000000000000000001000000000000000000000000000000000000000000000000000000001 = 3 333667  cdot mbox{BIG} 
20769187434139310658237173392736257 = 73  cdot mbox{BIG} 
57  GCD 1  m  is ODD  

58
100000000000000000000000000000000000000000000000000000000010000000000000000000000000000000000000000000000000000000001 = 3 7 13 37 4003 72559  cdot mbox{BIG} 
83076749736557242344718317419233281 = 3 7 4177  cdot mbox{BIG} 
58  GCD 21

59
10000000000000000000000000000000000000000000000000000000000100000000000000000000000000000000000000000000000000000000001 = 3 37 3187  cdot mbox{BIG} 
332306998946228968802412517373509633 = 7 184081  cdot mbox{BIG} 
59  GCD 1  m  is ODD  

60
1000000000000000000000000000000000000000000000000000000000001000000000000000000000000000000000000000000000000000000000001 = 3 19 181 29611 52579 238681 333667  cdot mbox{BIG} 
1329227995784915874056728564887191553 = 3 19 37 73 109 181 631 23311 54001  cdot mbox{BIG} 
60  GCD 10317

61
100000000000000000000000000000000000000000000000000000000000010000000000000000000000000000000000000000000000000000000000001 = 3 37  cdot mbox{BIG} 
5316911983139663493921071250335072257 = 7 367 55633  cdot mbox{BIG} 
61  GCD 1  m  is ODD  

62
10000000000000000000000000000000000000000000000000000000000000100000000000000000000000000000000000000000000000000000000000001 = 3 7 13 37 373 44641  cdot mbox{BIG} 
21267647932558653971072598982912901121 = 3 7  cdot mbox{BIG} 
62  GCD 21

63
1000000000000000000000000000000000000000000000000000000000000001000000000000000000000000000000000000000000000000000000000000001 = 3 757  cdot mbox{BIG} 
85070591730234615875067023894796828673 = 262657  cdot mbox{BIG} 
63  GCD 1  m  is ODD  

64
100000000000000000000000000000000000000000000000000000000000000010000000000000000000000000000000000000000000000000000000000000001 = 3 7 13 37 97 193 769 9901 206209  cdot mbox{BIG} 
340282366920938463481821351505477763073 = 3 7 13 97 193 241 673  cdot mbox{BIG} 
64  GCD 5110833

65
10000000000000000000000000000000000000000000000000000000000000000100000000000000000000000000000000000000000000000000000000000000001 = 3 31 37 1951  cdot mbox{BIG} 
1361129467683753853890391917874491949057 = 7 79 151 121369  cdot mbox{BIG} 
65  GCD 1  m  is ODD  

66
1000000000000000000000000000000000000000000000000000000000000000001000000000000000000000000000000000000000000000000000000000000000001 = 3 19 199 397 34849 52579 333667  cdot mbox{BIG} 
5444517870735015415487780695203129589761 = 3 19 73 199 5347 153649  cdot mbox{BIG} 
66  GCD 11343

67
100000000000000000000000000000000000000000000000000000000000000000010000000000000000000000000000000000000000000000000000000000000000001 = 3 37 1609  cdot mbox{BIG} 
21778071482940061661803548828222841946113 = 7 1609 22111  cdot mbox{BIG} 
67  GCD 1609  m  is ODD  

68
10000000000000000000000000000000000000000000000000000000000000000000100000000000000000000000000000000000000000000000000000000000000000001 = 3 7 13 37 409 613 3061 9901 210631  cdot mbox{BIG} 
87112285931760246646919047407712014958593 = 3 7 13 103 307 409 2143 2857 3061 6529 11119 13669  1326700741
68  GCD 341782077

69
1000000000000000000000000000000000000000000000000000000000000000000001000000000000000000000000000000000000000000000000000000000000000000001 = 3 333667  cdot mbox{BIG} 
348449143727040986587085893820489354182657 = 73 79903  cdot mbox{BIG} 
69  GCD 1  m  is ODD  

70
100000000000000000000000000000000000000000000000000000000000000000000010000000000000000000000000000000000000000000000000000000000000000000001 = 3 7^2 13 31 37 43 127 211 241 1933 2161 2689 459691  cdot mbox{BIG} 
1393796574908163946347162983661240005427201 = 3 7^2 151 211 331 337 5419 29191 106681 152041 664441  1564921
70  GCD 31017

71
10000000000000000000000000000000000000000000000000000000000000000000000100000000000000000000000000000000000000000000000000000000000000000000001 = 3 37 853  cdot mbox{BIG} 
5575186299632655785386290751403525199101953 = 7 66457  cdot mbox{BIG} 
71  GCD 1  m  is ODD  

72
1000000000000000000000000000000000000000000000000000000000000000000000001000000000000000000000000000000000000000000000000000000000000000000000001 = 3 109 757 153469  cdot mbox{BIG} 
22300745198530623141540440639131231151194113 = 3 87211 246241 262657 279073  cdot mbox{BIG} 
72  GCD 3

73
100000000000000000000000000000000000000000000000000000000000000000000000010000000000000000000000000000000000000000000000000000000000000000000000001 = 3 37 439  cdot mbox{BIG} 
89202980794122492566152317823559185314349057 = 7 3943  cdot mbox{BIG} 
73  GCD 1  m  is ODD  

74
10000000000000000000000000000000000000000000000000000000000000000000000000100000000000000000000000000000000000000000000000000000000000000000000000001 = 3 7 13 37^2 223 4663  cdot mbox{BIG} 
356811923176489970264590381828305262676541441 = 3 7 3331 17539 321679  cdot mbox{BIG} 
74  GCD 21

75
1000000000000000000000000000000000000000000000000000000000000000000000000001000000000000000000000000000000000000000000000000000000000000000000000000001 = 3 238681 333667  cdot mbox{BIG} 
1427247692705959881058323748381358093544456193 = 73 631 23311 115201 617401  cdot mbox{BIG} 
75  GCD 1  m  is ODD  

76
100000000000000000000000000000000000000000000000000000000000000000000000000010000000000000000000000000000000000000000000000000000000000000000000000000001 = 3 7 13 37 229 2281 4789 9901 21319  cdot mbox{BIG} 
5708990770823839524233219435661706459854405633 = 3 7 13 571 32377 131101 160969  cdot mbox{BIG} 
76  GCD 273

77
10000000000000000000000000000000000000000000000000000000000000000000000000000100000000000000000000000000000000000000000000000000000000000000000000000000001 = 3 37 43 67 1933 2311  cdot mbox{BIG} 
22835963083295358096932726626919374010770784257 = 7^2 337 463 599479  cdot mbox{BIG} 
77  GCD 1  m  is ODD  

78
1000000000000000000000000000000000000000000000000000000000000000000000000000001000000000000000000000000000000000000000000000000000000000000000000000000000001 = 3 19 52579 333667 461917  cdot mbox{BIG} 
91343852333181432387730604276222592385789460481 = 3 19 73 937 6553 86113  cdot mbox{BIG} 
78  GCD 57

79
100000000000000000000000000000000000000000000000000000000000000000000000000000010000000000000000000000000000000000000000000000000000000000000000000000000000001 = 3 37  cdot mbox{BIG} 
365375409332725729550921812641980562228570488833 = 7 1423 49297  cdot mbox{BIG} 
79  GCD 1  m  is ODD  

80
10000000000000000000000000000000000000000000000000000000000000000000000000000000100000000000000000000000000000000000000000000000000000000000000000000000000000001 = 3 7 13 31 37 61 211 241 2161 9901  cdot mbox{BIG} 
1461501637330902918203686041642102634285107249153 = 3 7 13 61 97 151 241 331 673 1321  cdot mbox{BIG} 
80  GCD 4013373

81
1000000000000000000000000000000000000000000000000000000000000000000000000000000001000000000000000000000000000000000000000000000000000000000000000000000000000000001 = 3  cdot mbox{BIG} 
5846006549323611672814741748716771307882079584257 = 487  cdot mbox{BIG} 
81  GCD 1  m  is ODD  

82
100000000000000000000000000000000000000000000000000000000000000000000000000000000010000000000000000000000000000000000000000000000000000000000000000000000000000000001 = 3 7 13 37 739 148339  cdot mbox{BIG} 
23384026197294446691258962159163806773011619512321 = 3 7 739 165313  cdot mbox{BIG} 
82  GCD 15519

83
10000000000000000000000000000000000000000000000000000000000000000000000000000000000100000000000000000000000000000000000000000000000000000000000000000000000000000000001 = 3 37 4483  cdot mbox{BIG} 
93536104789177786765035838965248670175013080399873 = 7  cdot mbox{BIG} 
83  GCD 1  m  is ODD  

84
1000000000000000000000000000000000000000000000000000000000000000000000000000000000001000000000000000000000000000000000000000000000000000000000000000000000000000000000001 = 3 19 1009 10837 23311 45613 52579 333667  cdot mbox{BIG} 
374144419156711147060143336518181566865985526300673 = 3 19 37 73 109 92737 649657  cdot mbox{BIG} 
84  GCD 57

85
100000000000000000000000000000000000000000000000000000000000000000000000000000000000010000000000000000000000000000000000000000000000000000000000000000000000000000000000001 = 3 31 37 613 210631  cdot mbox{BIG} 
1496577676626844588240573307387100039795808514605057 = 7 103 151 2143 11119 106591 949111  cdot mbox{BIG} 
85  GCD 1  m  is ODD  

86
10000000000000000000000000000000000000000000000000000000000000000000000000000000000000100000000000000000000000000000000000000000000000000000000000000000000000000000000000001 = 3 7 13 37  cdot mbox{BIG} 
5986310706507378352962293152177147703846966877224961 = 3 7 1033  cdot mbox{BIG} 
86  GCD 21

87
1000000000000000000000000000000000000000000000000000000000000000000000000000000000000001000000000000000000000000000000000000000000000000000000000000000000000000000000000000001 = 3 523 333667 670249  cdot mbox{BIG} 
23945242826029513411849172453966085904715333146509313 = 73  cdot mbox{BIG} 
87  GCD 1  m  is ODD  

88
100000000000000000000000000000000000000000000000000000000000000000000000000000000000000010000000000000000000000000000000000000000000000000000000000000000000000000000000000000001 = 3 7 13 37 67 2377 9901 16369 432961  cdot mbox{BIG} 
95780971304118053647396689506379333797516263861256193 = 3 7 13 67 241 7393 20857 312709 599479  cdot mbox{BIG} 
88  GCD 18291

89
10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000100000000000000000000000000000000000000000000000000000000000000000000000000000000000000001 = 3 37 9613  cdot mbox{BIG} 
383123885216472214589586757406547315547374917995462657 = 7  cdot mbox{BIG} 
89  GCD 1  m  is ODD  

90
1000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001 = 3 757 6481  cdot mbox{BIG} 
1532495540865888858358347028388249222904119397082726401 = 3 271 811 15121 87211 262657 348031  cdot mbox{BIG} 
90  GCD 3

91
100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000010000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001 = 3 37 43 1093 1933  cdot mbox{BIG} 
6129982163463555433433388111077116813045717038532657153 = 7^2 79 337 121369  cdot mbox{BIG} 
91  GCD 1  m  is ODD  

92
10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001 = 3 7 13 37 277 829 9901 31051  cdot mbox{BIG} 
24519928653854221733733552439356707095041347054534131713 = 3 7 13 139  cdot mbox{BIG} 
92  GCD 273

93
1000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001 = 3 333667  cdot mbox{BIG} 
98079714615416886934934209747523308065882346018943533057 = 73 16183 34039  cdot mbox{BIG} 
93  GCD 1  m  is ODD  

94
100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000010000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001 = 3 7 13 37 283 344887  cdot mbox{BIG} 
392318858461667547739736838970286191634963299677388144641 = 3 7  cdot mbox{BIG} 
94  GCD 21

95
10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001 = 3 31 37 21319  cdot mbox{BIG} 
1569275433846670190958947355841530685282721029912780603393 = 7 151 32377  cdot mbox{BIG} 
95  GCD 1  m  is ODD  

96
1000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001 = 3 19 3169 8929 13249 52579 98641 333667  cdot mbox{BIG} 
6277101735386680763835789423286894578616619782057578463233 = 3 19 37 73 109 433 577 1153 6337 38737  cdot mbox{BIG} 
96  GCD 57

97
100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000010000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001 = 3 37 1747 18043  cdot mbox{BIG} 
25108406941546723055343157692989121989437950453043225952257 = 7 272959  cdot mbox{BIG} 
97  GCD 1  m  is ODD  

98
10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001 = 3 7^3 13 37 43 127 1933 2689 63799 459691  cdot mbox{BIG} 
100433627766186892221372630771639575307694744461798728007681 = 3 7^3 337 5419 748819  cdot mbox{BIG} 
98  GCD 1029

99
1000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001 = 3 757 55243 198397  cdot mbox{BIG} 
401734511064747568885490523085924475930664863146446560428033 = 262657  cdot mbox{BIG} 
99  GCD 1  m  is ODD  

100
100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000010000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001 = 3 7 13 31 37 61 151 211 241 601 2161 4201 9901 261301  cdot mbox{BIG} 
1606938044258990275541962092342430253122431223184289538506753 = 3 7 13 61 151 331 1201 1321 63901 100801  cdot mbox{BIG} 
100  GCD 2514603