When does $2^{2m}+2^m+1$ divide $10^{2m}+10^m+1$?
partial answer: using my comment we get:
$\begin{align} \\
2^{2m}+2^m+1 &\mid 2^{2m}5^{2m}+2^m5^m+1 \\
&\implies 2^{2m}+2^m+1 \mid 2^{2m}(5^{2m}-1)+2^m(5^m-1) \\
&\implies 2^{2m}+2^m+1 \mid 2^m\big({5^{2m}-1\over 4}\big)+\big({5^m-1\over 4}\big) \\
&\implies 2^m(2^m+1)+1 \mid 2^m\big({5^{2m}-1\over 4}\big)+\big({5^m-1\over 4}\big)
\end{align}$
hope this helps simplify things. edit: the fractions are the base 5 repunits.
LATE Edit: if M is divisible by totient function values you can also use Euler's theorem to prove that both are 3 mod what ever numbers give those totients. then it's about when two numbers on arithmetic progressions divide each other.
Partial result covering approximately two thirds of all the numbers $m$.
I am looking for a covering set of primes. The idea is that whenever we find a prime $p$ such that $p\mid 4^m+2^m+1$ but $p\nmid 100^m+10^m+1$ we have shown that $4^m+2^m+1\nmid 100^m+10^m+1$. When hunting for suitable primes $p$ we can simply look at the factorizations of these numbers. For $p>3$ to be a factor of $k^{2m}+k^m+1=(k^{3m}-1)/(k^m-1)$ we see that $k^m$ has to be a primitive third root of unity modulo $p$. We obviously apply this to both $k=2$ and $k=10$.
To get those primitive third roots of unity we need $\ell=\operatorname{ord}_p(2)$ to be divisible by three. If $\ell=3u$ we then see that $p\mid 4^m+2^m+1$ if and only if $m\equiv u,2u\pmod{3u}$. But, if $t=\operatorname{ord}_p(10)$ is also divisible by three, say $t=3v$, we also have that $p\mid 100^m+10^m+1$ whenever $m\equiv v,2v\pmod{3v}$. Anyway, this allows to conclude that $m$ cannot be in certain residue classes modulo $3w$ where $w=\operatorname{lcm}(u,v)$. Observe that if $t$ is not a multiple of three, then $100^m+10^m+1$ is never divisible by $p$, because $10^m$ is then never a primitive third root of unity.
Getting our hands dirty we see that:
- With $p=7$ we have $\ell=3$ and $t=6$, both divisible by three. This means that when $m\equiv1,5\pmod6$ we have $7\mid 4^m+2^m+1$ and $7\nmid 100^m+10^m+1$.
- With $p=73\mid 2^9-1$ we have $\ell=9=3\cdot3$ and $t=8$. This means that $73\mid 4^m+2^m+1$ iff $m\equiv3,6\pmod9$. But $100^m+10^m+1$ is never divisible by $73$ so we have covered these values of $m$. At this point we have excluded $10$ out of $18$ residue classes modulo $18$, so we have covered more than half the integers.
- The prime $p=13$ is a factor of $2^{12}-1$ and $\ell=12$. But here $t=6$ is a factor of $\ell$. This is bad news for us. No values of $m$ can be excluded. For if $m\equiv4,8\pmod{12}$ then automatically $m\equiv2,4\pmod{6}$. In these cases $13$ is a common factor of both $4^m+2^m+1$ and $100^m+10^m+1$. The same thing happens whenever $2$ is a primitive root modulo $p$. We really need to look for non-primitive cases.
- With $p=151\mid 2^{15}-1$ we have $\ell=15=3\cdot5$. Here $t=75=3\cdot25$. This means that we can rule out $m$, if $m\equiv5,10\pmod{15}$ and $m\not\equiv25,50\pmod{75}$.
- With $p=241\mid 2^{24}-1$ we have $\ell=24=3\cdot8$. Here $t=30=3\cdot10$. We can rule ouf $m$, if $m\equiv8,16\pmod{24}$ and $m\not\equiv10,20\pmod{30}$.
At this point we have ruled out $1216$ residue classes modulo $1800$, so we reached the two thirds point. That modulus $1800$ is the least common multiple of $6,9,75,24$ and $30$ that all appeared above. Recalling that $m=0,2,4$ are solutions the smallest values of $m$ that we have not yet covered are $m=9,14,18,22$. Stopping here for now.
up to 100, the odd $m$ for which the 2 number and the 10 number are not coprime are $m=25,33,67,$ while these gcd's are still quite small compared with the raw numbers.
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