mersenneforum.org  

Go Back   mersenneforum.org > New To GIMPS? Start Here! > Homework Help

Reply
 
Thread Tools
Old 2022-06-23, 16:53   #1
bur
 
bur's Avatar
 
Aug 2020
79*6581e-4;3*2539e-3

7×83 Posts
Default Peculiar divisors of k.10^n-1

I noticed that 3 * 10^272 - 1 / 13 = 230769...23076923. This periodic appearance in base 10 shows for all exponents where the number is divisible by 13, i.e. 3 * 10^32 - 1 / 13 = 23076923076923076923076923076923.

It is also not restricted to k= 3, as 9 * 10^16 - 1 / 13 = 6923076923076923 or 12 * 10^13 - 1 / 13 = 923076923076923 or 35 * 10^16 - 1 / 13 = 26923076923076923 etc.

It's not actually homework or an exercise, but if someone could give a hint instead of the full explanation, I'd be happy to try and figure the rest out myself.
bur is offline   Reply With Quote
Old 2022-06-23, 20:58   #2
xilman
Bamboozled!
 
xilman's Avatar
 
"๐’‰บ๐’ŒŒ๐’‡ท๐’†ท๐’€ญ"
May 2003
Down not across

11,423 Posts
Default

Quote:
Originally Posted by bur View Post
I noticed that 3 * 10^272 - 1 / 13 = 230769...23076923. This periodic appearance in base 10 shows for all exponents where the number is divisible by 13, i.e. 3 * 10^32 - 1 / 13 = 23076923076923076923076923076923.

It is also not restricted to k= 3, as 9 * 10^16 - 1 / 13 = 6923076923076923 or 12 * 10^13 - 1 / 13 = 923076923076923 or 35 * 10^16 - 1 / 13 = 26923076923076923 etc.

It's not actually homework or an exercise, but if someone could give a hint instead of the full explanation, I'd be happy to try and figure the rest out myself.
Hint: use the long division algorithm you (should have) learned at school and pay attention to the remainders.
xilman is online now   Reply With Quote
Old 2022-06-24, 07:14   #3
bur
 
bur's Avatar
 
Aug 2020
79*6581e-4;3*2539e-3

7×83 Posts
Default

Ah ok, so it's that simple... :D I was spending too much time analyzing the values of k Mod 13 and their order. Thanks.
bur is offline   Reply With Quote
Old 2022-06-24, 07:54   #4
Batalov
 
Batalov's Avatar
 
"Serge"
Mar 2008
Phi(4,2^7658614+1)/2

990210 Posts
Thumbs up

This is a good organoleptic entrance into the concept of unique primes. You have discovered it by experimentation. That's good.

You have just found that 1/13 is periodic with period of six. And so is 1/7, also periodic with period of six.
As a result both 7 and 13 are not unique primes.

Quote:
Originally Posted by https://primes.utm.edu/top20/page.php?id=62
The reciprocal of every prime p (other than two and five) has a period, that is the decimal expansion of 1/p repeats in blocks of some set length (see the period of a decimal expansion). This is called the period of the prime p.

Samuel Yates defined a unique prime (or unique period prime) to be a prime which has a period that it shares with no other prime. For example: 3, 11, 37, and 101 are the only primes with periods one, two, three, and four respectively--so they are unique primes. But 41 and 271 both have period five, 7 and 13 both have period six, 239 and 4649 both have period seven, and each of 353, 449, 641, 1409, and 69857 have period thirty-two, showing that these primes are not unique primes.
Batalov is offline   Reply With Quote
Old 2022-06-24, 12:30   #5
bur
 
bur's Avatar
 
Aug 2020
79*6581e-4;3*2539e-3

7·83 Posts
Default

Thanks, actually, I saw your various unique primes at Caldwell's list before, so I knew that concept, but didn't connect it to this phenomenon.

In hindsight this obviously occurs for every prime divisor of these numbers. I just didn't notice it because the period is much longer.
bur is offline   Reply With Quote
Old 2022-06-29, 10:19   #6
MattcAnderson
 
MattcAnderson's Avatar
 
"Matthew Anderson"
Dec 2010
Oregon, USA

5·233 Posts
Smile hi

My project - prime constellations and k-tuples
MattcAnderson is offline   Reply With Quote
Old 2022-06-29, 18:11   #7
sweety439
 
"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36

DFA16 Posts
Default

Quote:
Originally Posted by Batalov View Post
As a result both 7 and 13 are not unique primes.
7 and 13 are unique primes in base 2

7 is the only prime with period length 3 in base 2, and 13 is the only prime with period length 12 in base 2

see factorization of Phi(n,2)
sweety439 is offline   Reply With Quote
Reply

Thread Tools


Similar Threads
Thread Thread Starter Forum Replies Last Post
Peculiar activity in the 1M range... petrw1 PrimeNet 5 2011-01-14 18:01
Sum of two squares: Peculiar property Raman Puzzles 13 2010-02-13 05:25
Peculiar behaviour of prime numbers ..... spkarra Math 8 2009-07-20 22:47
Peculiar Case of Benjamin Button is AWESOME!!!!!!! jasong jasong 7 2009-01-01 00:50
A particularly peculiar problem fivemack Puzzles 7 2008-11-14 07:56

All times are UTC. The time now is 00:51.


Sun Aug 14 00:51:10 UTC 2022 up 37 days, 19:38, 2 users, load averages: 0.99, 1.16, 1.08

Powered by vBulletin® Version 3.8.11
Copyright ©2000 - 2022, Jelsoft Enterprises Ltd.

This forum has received and complied with 0 (zero) government requests for information.

Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation.
A copy of the license is included in the FAQ.

โ‰  ยฑ โˆ“ รท ร— ยท โˆ’ โˆš โ€ฐ โŠ— โŠ• โŠ– โŠ˜ โŠ™ โ‰ค โ‰ฅ โ‰ฆ โ‰ง โ‰จ โ‰ฉ โ‰บ โ‰ป โ‰ผ โ‰ฝ โŠ โŠ โŠ‘ โŠ’ ยฒ ยณ ยฐ
โˆ  โˆŸ ยฐ โ‰… ~ โ€– โŸ‚ โซ›
โ‰ก โ‰œ โ‰ˆ โˆ โˆž โ‰ช โ‰ซ โŒŠโŒ‹ โŒˆโŒ‰ โˆ˜ โˆ โˆ โˆ‘ โˆง โˆจ โˆฉ โˆช โจ€ โŠ• โŠ— ๐–• ๐–– ๐–— โŠฒ โŠณ
โˆ… โˆ– โˆ โ†ฆ โ†ฃ โˆฉ โˆช โŠ† โŠ‚ โŠ„ โŠŠ โŠ‡ โŠƒ โŠ… โŠ‹ โŠ– โˆˆ โˆ‰ โˆ‹ โˆŒ โ„• โ„ค โ„š โ„ โ„‚ โ„ต โ„ถ โ„ท โ„ธ ๐“Ÿ
ยฌ โˆจ โˆง โŠ• โ†’ โ† โ‡’ โ‡ โ‡” โˆ€ โˆƒ โˆ„ โˆด โˆต โŠค โŠฅ โŠข โŠจ โซค โŠฃ โ€ฆ โ‹ฏ โ‹ฎ โ‹ฐ โ‹ฑ
โˆซ โˆฌ โˆญ โˆฎ โˆฏ โˆฐ โˆ‡ โˆ† ฮด โˆ‚ โ„ฑ โ„’ โ„“
๐›ข๐›ผ ๐›ฃ๐›ฝ ๐›ค๐›พ ๐›ฅ๐›ฟ ๐›ฆ๐œ€๐œ– ๐›ง๐œ ๐›จ๐œ‚ ๐›ฉ๐œƒ๐œ— ๐›ช๐œ„ ๐›ซ๐œ… ๐›ฌ๐œ† ๐›ญ๐œ‡ ๐›ฎ๐œˆ ๐›ฏ๐œ‰ ๐›ฐ๐œŠ ๐›ฑ๐œ‹ ๐›ฒ๐œŒ ๐›ด๐œŽ๐œ ๐›ต๐œ ๐›ถ๐œ ๐›ท๐œ™๐œ‘ ๐›ธ๐œ’ ๐›น๐œ“ ๐›บ๐œ”