
Number Theory Number Theory Math Forum 
 LinkBack  Thread Tools  Display Modes 
August 11th, 2016, 07:09 PM  #1 
Newbie Joined: Aug 2016 From: Hong Kong Posts: 5 Thanks: 0  All primes are in form of 10nk+c
Prove: For positive integer n, and any positive integer c that 0 < c < 10n and c is coprime with 10n, all primes larger than 10n can be expressed in form of 10nk+c where k is a positive integer. Remark: c may have various values. For example, when n=2, possible values of c are: 1,3,7,9,11,13,17,19 
August 11th, 2016, 11:53 PM  #2 
Math Team Joined: Nov 2014 From: Australia Posts: 689 Thanks: 244 
Let $m = nk$. Then setting $n = 1$, it is clear that $m$ can be any natural number. It is a consequence of the division algorithm that any positive number (including primes) can be represented by $10m + c$, where $0 < c < 10$. This proves the result. 
August 12th, 2016, 08:31 AM  #3 
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,958 Thanks: 1146 Math Focus: Elementary mathematics and beyond 
But if $c$ is coprime to $10n$ how would you represent, say, 12?

August 12th, 2016, 08:59 AM  #4 
Global Moderator Joined: Dec 2006 Posts: 20,926 Thanks: 2205 
If $c$ isn't coprime to $10n$, $10n + c$ isn't a prime.

August 12th, 2016, 03:26 PM  #5 
Math Team Joined: Nov 2014 From: Australia Posts: 689 Thanks: 244 
Ah. I didn't read the question very well it seems. It is still trivial however, since 1, 3, 7 and 9 are all coprime to $10n$. This means that all odd numbers not ending in 5 (and thus not divisible by 5) can be represented in this way. 

Tags 
10nk, coprime, form, primes 
Thread Tools  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
Conjecture about primes of the form 2^k1  mobel  Number Theory  21  October 16th, 2015 07:45 AM 
primes and twin primes: Number between powers of 10  caters  Number Theory  67  March 19th, 2014 04:32 PM 
Conjecture about primes of a special form  Sebastian Garth  Number Theory  9  November 22nd, 2013 02:38 PM 
Help with an infinite primes of the form proof  WheepWhoop  Number Theory  7  October 20th, 2011 05:09 PM 
Are there infinitely many primes of the form (n!)/2+1 ?  johnmath  Number Theory  8  April 29th, 2011 08:45 AM 