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 2013-02-12, 16:17 #1 Nick     Dec 2012 The Netherlands 22·5·79 Posts 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
 2013-02-13, 06:18 #2 LaurV Romulan Interpreter     Jun 2011 Thailand 100011110000012 Posts [URL="http://oeis.org/A003226"]here is the link[/URL] Last fiddled with by LaurV on 2013-02-13 at 06:20 Reason: grrr, I did not foresee that the blue color of the link makes the test visible even when you use spoilers...
2013-02-13, 07:56   #3
Mr. P-1

Jun 2003

100100100012 Posts

Quote:
 Originally Posted by Nick 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?
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

 2013-02-13, 08:08 #4 Batalov     "Serge" Mar 2008 Phi(4,2^7658614+1)/2 2·11·421 Posts There was a nice problem on projecteuler relevant to this thread.
 2013-02-13, 08:52 #5 LaurV Romulan Interpreter     Jun 2011 Thailand 34·113 Posts 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 32 ms. 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 31 ms. gp > Last fiddled with by LaurV on 2013-02-13 at 09:01
 2013-02-13, 15:45 #6 Raman Noodles     "Mr. Tuch" Dec 2007 Chennai, India 125710 Posts 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. What patterns? - guess please! (1) (a) 3 53 753 0753 60753 660753 ... (b) 7 77 477 6477 46477 446477 ... (c) 1 71 471 8471 88471 288471 ... Hint: See what their own cubes end with! (2) (a) ...761... ...2761... ...32761... ...932761... ...3932761... (b) ...6403... ...56403... ...756403... ...7756403... (c) ...23... ...223... ...4223... ...34223... ...534223... ...7534223... 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].
 2013-02-13, 16:24 #7 fivemack (loop (#_fork))     Feb 2006 Cambridge, England 143538 Posts 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 )
 2013-02-13, 16:39 #8 fivemack (loop (#_fork))     Feb 2006 Cambridge, England 637910 Posts 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.
2013-02-13, 16:45   #9
Raman
Noodles

"Mr. Tuch"
Dec 2007
Chennai, India

3×419 Posts

Quote:
 Originally Posted by fivemack 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.
x1 = 7
x2 = 87
x3 = 587
x4 = 4587
x5 = 34587
x6 = 134587
x7 = 5134587
x8 = 95134587
x9 = 895134587
x10 = 7895134587
x11 = 87895134587
x12 = 087895134587
x13 = 0087895134587
x14 = 50087895134587

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

Last fiddled with by Raman on 2013-02-13 at 16:46

 2013-02-13, 17:17 #10 Nick     Dec 2012 The Netherlands 62C16 Posts 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 $\mathbb{Z}/10^n\mathbb{Z}$ is isomorphic to $\mathbb{Z}/5^n\mathbb{Z}\times\mathbb{Z}/2^n\mathbb{Z}$. 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. Last fiddled with by Nick on 2013-02-13 at 17:19

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