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-   -   Playing with decimal representation (https://www.mersenneforum.org/showthread.php?t=17785)

 Nick 2013-02-12 16:17

Playing with decimal representation

Nothing deep here, just a light-hearted puzzle.
Find the next number (i.e. its first digit) in the following sequence:
5
25
625
0625
90625
890625
2890625
12890625
212890625
8212890625
18212890625
918212890625
...
2166509580863811000557423423230896109004106619977392256259918212890625
and then?

Now do the same for:
6
76
376
...
7833490419136188999442576576769103890995893380022607743740081787109376

 LaurV 2013-02-13 06:18

 Mr. P-1 2013-02-13 07:56

[QUOTE=Nick;329105]Nothing deep here, just a light-hearted puzzle.
Find the next number (i.e. its first digit) in the following sequence:
5
25
625
0625
90625
890625
2890625
12890625
212890625
8212890625
18212890625
918212890625
...
2166509580863811000557423423230896109004106619977392256259918212890625
and then?[/quote]

The last n+1 digits of 5^(2^n) starting with n=0

Now do the same for:
6
76
376

...
7833490419136188999442576576769103890995893380022607743740081787109376[/QUOTE]

The last n+1 digits of 376^(2^n) starting with n=0

 Batalov 2013-02-13 08:08

There was a nice [URL="http://projecteuler.net/problem=284"]problem[/URL] on projecteuler relevant to this thread.

 LaurV 2013-02-13 08:52

ok, got a bit of free time during lunch break:

[edit: scroll down, till the red texts]
[edit 2: all the "log" stuff is no needed, it is just to display the number of digits, I was lazy to use counters]

[CODE]gp > n=5; log10=log(10); s=1; while(n<10^100, if(n^2%10^(cntdig=log(n)\log10+1)==n, print(cntdig": "n); s=10^cntdig); n+=s)
1: 5
2: 25
3: 625
5: 90625
6: 890625
7: 2890625
8: 12890625
9: 212890625
10: 8212890625
11: 18212890625
12: 918212890625
13: 9918212890625
14: 59918212890625
15: 259918212890625
16: 6259918212890625
17: 56259918212890625
18: 256259918212890625
19: 2256259918212890625
20: 92256259918212890625
21: 392256259918212890625
22: 7392256259918212890625
23: 77392256259918212890625
24: 977392256259918212890625
25: 9977392256259918212890625
26: 19977392256259918212890625
27: 619977392256259918212890625
28: 6619977392256259918212890625
30: 106619977392256259918212890625
31: 4106619977392256259918212890625
34: 9004106619977392256259918212890625
36: 109004106619977392256259918212890625
37: 6109004106619977392256259918212890625
38: 96109004106619977392256259918212890625
39: 896109004106619977392256259918212890625
41: 30896109004106619977392256259918212890625
42: 230896109004106619977392256259918212890625
43: 3230896109004106619977392256259918212890625
44: 23230896109004106619977392256259918212890625
45: 423230896109004106619977392256259918212890625
46: 3423230896109004106619977392256259918212890625
47: 23423230896109004106619977392256259918212890625
48: 423423230896109004106619977392256259918212890625
49: 7423423230896109004106619977392256259918212890625
50: 57423423230896109004106619977392256259918212890625
51: 557423423230896109004106619977392256259918212890625
55: 1000557423423230896109004106619977392256259918212890625
56: 11000557423423230896109004106619977392256259918212890625
57: 811000557423423230896109004106619977392256259918212890625
58: 3811000557423423230896109004106619977392256259918212890625
59: 63811000557423423230896109004106619977392256259918212890625
60: 863811000557423423230896109004106619977392256259918212890625
62: 80863811000557423423230896109004106619977392256259918212890625
63: 580863811000557423423230896109004106619977392256259918212890625
64: 9580863811000557423423230896109004106619977392256259918212890625
66: 509580863811000557423423230896109004106619977392256259918212890625
67: 6509580863811000557423423230896109004106619977392256259918212890625
68: 66509580863811000557423423230896109004106619977392256259918212890625
69: 166509580863811000557423423230896109004106619977392256259918212890625
70: 2166509580863811000557423423230896109004106619977392256259918212890625
71: 62166509580863811000557423423230896109004106619977392256259918212890625
74: 90062166509580863811000557423423230896109004106619977392256259918212890625
75: 890062166509580863811000557423423230896109004106619977392256259918212890625
76: 9890062166509580863811000557423423230896109004106619977392256259918212890625
78: 509890062166509580863811000557423423230896109004106619977392256259918212890625
79: 8509890062166509580863811000557423423230896109004106619977392256259918212890625
80: 38509890062166509580863811000557423423230896109004106619977392256259918212890625
81: 938509890062166509580863811000557423423230896109004106619977392256259918212890625
82: 2938509890062166509580863811000557423423230896109004106619977392256259918212890625
84: 802938509890062166509580863811000557423423230896109004106619977392256259918212890625
85: 9802938509890062166509580863811000557423423230896109004106619977392256259918212890625
86: 69802938509890062166509580863811000557423423230896109004106619977392256259918212890625
87: 169802938509890062166509580863811000557423423230896109004106619977392256259918212890625
88: 8169802938509890062166509580863811000557423423230896109004106619977392256259918212890625
90: 108169802938509890062166509580863811000557423423230896109004106619977392256259918212890625
91: 9108169802938509890062166509580863811000557423423230896109004106619977392256259918212890625
92: 19108169802938509890062166509580863811000557423423230896109004106619977392256259918212890625
93: 319108169802938509890062166509580863811000557423423230896109004106619977392256259918212890625
94: 7319108169802938509890062166509580863811000557423423230896109004106619977392256259918212890625
97: 3007319108169802938509890062166509580863811000557423423230896109004106619977392256259918212890625
98: 53007319108169802938509890062166509580863811000557423423230896109004106619977392256259918212890625
99: 953007319108169802938509890062166509580863811000557423423230896109004106619977392256259918212890625
100: 3953007319108169802938509890062166509580863811000557423423230896109004106619977392256259918212890625
gp > ##
*** last result computed in [B][COLOR=Red]32 ms[/COLOR][/B].
gp > n=6; log10=log(10); s=1; while(n<10^100, if(n^2%10^(cntdig=log(n)\log10+1)==n, print(cntdig": "n); s=10^cntdig); n+=s)
1: 6
2: 76
3: 376
4: 9376
6: 109376
7: 7109376
8: 87109376
9: 787109376
10: 1787109376
11: 81787109376
14: 40081787109376
15: 740081787109376
16: 3740081787109376
17: 43740081787109376
18: 743740081787109376
19: 7743740081787109376
21: 607743740081787109376
22: 2607743740081787109376
23: 22607743740081787109376
26: 80022607743740081787109376
27: 380022607743740081787109376
28: 3380022607743740081787109376
29: 93380022607743740081787109376
30: 893380022607743740081787109376
31: 5893380022607743740081787109376
32: 95893380022607743740081787109376
33: 995893380022607743740081787109376
35: 90995893380022607743740081787109376
36: 890995893380022607743740081787109376
37: 3890995893380022607743740081787109376
39: 103890995893380022607743740081787109376
40: 9103890995893380022607743740081787109376
41: 69103890995893380022607743740081787109376
42: 769103890995893380022607743740081787109376
43: 6769103890995893380022607743740081787109376
44: 76769103890995893380022607743740081787109376
45: 576769103890995893380022607743740081787109376
46: 6576769103890995893380022607743740081787109376
47: 76576769103890995893380022607743740081787109376
48: 576576769103890995893380022607743740081787109376
49: 2576576769103890995893380022607743740081787109376
50: 42576576769103890995893380022607743740081787109376
51: 442576576769103890995893380022607743740081787109376
52: 9442576576769103890995893380022607743740081787109376
53: 99442576576769103890995893380022607743740081787109376
54: 999442576576769103890995893380022607743740081787109376
55: 8999442576576769103890995893380022607743740081787109376
56: 88999442576576769103890995893380022607743740081787109376
57: 188999442576576769103890995893380022607743740081787109376
58: 6188999442576576769103890995893380022607743740081787109376
59: 36188999442576576769103890995893380022607743740081787109376
60: 136188999442576576769103890995893380022607743740081787109376
61: 9136188999442576576769103890995893380022607743740081787109376
62: 19136188999442576576769103890995893380022607743740081787109376
63: 419136188999442576576769103890995893380022607743740081787109376
65: 90419136188999442576576769103890995893380022607743740081787109376
66: 490419136188999442576576769103890995893380022607743740081787109376
67: 3490419136188999442576576769103890995893380022607743740081787109376
68: 33490419136188999442576576769103890995893380022607743740081787109376
69: 833490419136188999442576576769103890995893380022607743740081787109376
70: 7833490419136188999442576576769103890995893380022607743740081787109376
71: 37833490419136188999442576576769103890995893380022607743740081787109376
72: 937833490419136188999442576576769103890995893380022607743740081787109376
73: 9937833490419136188999442576576769103890995893380022607743740081787109376
75: 109937833490419136188999442576576769103890995893380022607743740081787109376
77: 90109937833490419136188999442576576769103890995893380022607743740081787109376
78: 490109937833490419136188999442576576769103890995893380022607743740081787109376
79: 1490109937833490419136188999442576576769103890995893380022607743740081787109376
80: 61490109937833490419136188999442576576769103890995893380022607743740081787109376
82: 7061490109937833490419136188999442576576769103890995893380022607743740081787109376
83: 97061490109937833490419136188999442576576769103890995893380022607743740081787109376
84: 197061490109937833490419136188999442576576769103890995893380022607743740081787109376
86: 30197061490109937833490419136188999442576576769103890995893380022607743740081787109376
87: 830197061490109937833490419136188999442576576769103890995893380022607743740081787109376
88: 1830197061490109937833490419136188999442576576769103890995893380022607743740081787109376
89: 91830197061490109937833490419136188999442576576769103890995893380022607743740081787109376
90: 891830197061490109937833490419136188999442576576769103890995893380022607743740081787109376
92: 80891830197061490109937833490419136188999442576576769103890995893380022607743740081787109376
93: 680891830197061490109937833490419136188999442576576769103890995893380022607743740081787109376
94: 2680891830197061490109937833490419136188999442576576769103890995893380022607743740081787109376
95: 92680891830197061490109937833490419136188999442576576769103890995893380022607743740081787109376
96: 992680891830197061490109937833490419136188999442576576769103890995893380022607743740081787109376
97: 6992680891830197061490109937833490419136188999442576576769103890995893380022607743740081787109376
98: 46992680891830197061490109937833490419136188999442576576769103890995893380022607743740081787109376
100: 6046992680891830197061490109937833490419136188999442576576769103890995893380022607743740081787109376
gp > ##
*** last result computed in [B][COLOR=Red]31 ms[/COLOR][/B].
gp >[/CODE]

 Raman 2013-02-13 15:45

Okay. Enough is enough. These patterns are being quite popular enough.

Now, that if in case someone has got with enough strength, then he could be able to be
guessing into with these patterns.

(1) (a)
3
53
753
0753
60753
660753
...

(b)
7
77
477
6477
46477
446477
...

(c)
1
71
471
8471
88471
288471
...

[SPOILER]Hint: See what their own cubes end with![/SPOILER]

(2) (a)
...761...
...2761...
...32761...
...932761...
...3932761...

(b)
...6403...
...56403...
...756403...
...7756403...

(c)
...23...
...223...
...4223...
...34223...
...534223...
...7534223...

[SPOILER]Hint: Look at the last digits of 3[sup]5×10[sup]n+1[/sup][/sup],
7[sup]5×10[sup]n+1[/sup][/sup], 11[sup]5×10[sup]n+1[/sup][/sup].[/SPOILER]

 fivemack 2013-02-13 16:24

10-adic (to the extent that that makes sense) solutions to u^2=u

(they sum to 1, because u^2=u => (1-u)^2 = 1-2u+u^2 = 1-u )

 fivemack 2013-02-13 16:39

50087895134587 ...

(ah yes, the two initial ones are 0-mod-2 1-mod-5 and 0-mod-5 1-mod-2)

This is about the point where my PhD ran into the sands; these things are near enough to reals that you can contemplate integrating over them, but at that point my brain stopped.

 Raman 2013-02-13 16:45

[QUOTE=fivemack;329337]50087895134587 ...

(ah yes, the two initial ones are 0-mod-2 1-mod-5 and 0-mod-5 1-mod-2)

This is about the point where my PhD ran into the sands; these things are near enough to reals that you can contemplate integrating over them, but at that point my brain stopped.[/QUOTE]

x[SUB]1[/SUB] = 7
x[SUB]2[/SUB] = 87
x[SUB]3[/SUB] = 587
x[SUB]4[/SUB] = 4587
x[SUB]5[/SUB] = 34587
x[SUB]6[/SUB] = 134587
x[SUB]7[/SUB] = 5134587
x[SUB]8[/SUB] = 95134587
x[SUB]9[/SUB] = 895134587
x[SUB]10[/SUB] = 7895134587
x[SUB]11[/SUB] = 87895134587
x[SUB]12[/SUB] = 087895134587
x[SUB]13[/SUB] = 0087895134587
x[SUB]14[/SUB] = 50087895134587

[SPOILER]x[SUB]n[/SUB][SUP]3[/SUP] mod 10[SUP]n[/SUP] = 3[/SPOILER]

 Nick 2013-02-13 17:17

Wow, thanks for all the responses to the first puzzle, both answers and links.

Mathematically, the pattern arises because we use decimal notation and there are 2 distinct primes dividing 10 (it wouldn't work in hexadecimal, for example).
For any positive integer n the Chinese Remainder Theorem tells us that [TEX]\mathbb{Z}/10^n\mathbb{Z}[/TEX] is isomorphic to [TEX]\mathbb{Z}/5^n\mathbb{Z}\times\mathbb{Z}/2^n\mathbb{Z}[/TEX].

As fivemack pointed out, it follows that in the integers modulo 10ⁿ there are 4 idempotent elements (numbers equal to their own square), not just 0 and 1 (corresponding with the pairs (0,0) and (1,1)) but also the numbers corresponding with (1,0) and (0,1) via the isomorphism above.
Thus, for example, 625 is the unique integer modulo 1000 which is congruent to 0 mod 125 and 1 mod 8, while 376 is the unique integer modulo 1000 which is congruent to 1 mod 125 and 0 mod 8. And their sum 625+376 is congruent to 1 mod 1000 because (again as fivemack noted) if e is idempotent then so is 1-e.

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