My Math Forum  

Go Back   My Math Forum > College Math Forum > Number Theory

Number Theory Number Theory Math Forum

Thanks Tree5Thanks
  • 1 Post By greg1313
  • 2 Post By skipjack
  • 2 Post By Azzajazz
LinkBack Thread Tools Display Modes
August 11th, 2016, 07:09 PM   #1
Joined: Aug 2016
From: Hong Kong

Posts: 5
Thanks: 0

Question All primes are in form of 10nk+c


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
MarkRose is offline  
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.
Azzajazz is offline  
August 12th, 2016, 08:31 AM   #3
Global Moderator
greg1313's Avatar
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?
Thanks from Joppy
greg1313 is offline  
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.
Thanks from greg1313 and Joppy
skipjack is online now  
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.
Thanks from greg1313 and Joppy
Azzajazz is offline  

  My Math Forum > College Math Forum > Number Theory

10nk, coprime, form, primes

Thread Tools
Display Modes

Similar Threads
Thread Thread Starter Forum Replies Last Post
Conjecture about primes of the form 2^k-1 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

Copyright © 2019 My Math Forum. All rights reserved.