
Number Theory Number Theory Math Forum 
 LinkBack  Thread Tools  Display Modes 
October 2nd, 2010, 02:32 AM  #1 
Newbie Joined: Jun 2009 Posts: 14 Thanks: 0  2 elementary number theory problems.... problem 1: suppose is a prime number such that (P1)/4 and (P+1)/2 are also primes. Show that p = 13. problem 2: let be n numbers such that each is either or . If , then prove that divides . HELP! 
October 2nd, 2010, 09:36 PM  #2 
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 937 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Re: 2 elementary number theory problems....
For #1, work mod 4.

October 4th, 2010, 06:31 PM  #3 
Newbie Joined: Sep 2010 Posts: 17 Thanks: 0  Re: 2 elementary number theory problems....
If P, (P+1)/2 and (P1)/4 are prime numbers then P=4x+1, so our numbers are 4x+1, 2x+1 and x. Now, if x=3K then K must be 1, x=3. if x=3K+1 then 2x+1=2(3K+1)+1=3(2K+1). Then 2K+1 must be 1, so x=1 which is not prime. if x=3K+2 then 4x+1=4(3K+2)+1=3(4K+3). Then 4K+3 must be 1, and x isn't even an integer. So the only solution is x=3, P=13. ;] 
October 8th, 2010, 07:42 AM  #4 
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,697 Thanks: 977 Math Focus: Elementary mathematics and beyond  Re: 2 elementary number theory problems....
Clearly, p = 2, 3 are not solutions. Assume p is of the form 6n  1. p = 6n  1, (p  1)/4 = (3n  1)/2, (p + 1)/2 = 3n; 3n can never be prime so there are no primes of the form 6n  1 that are solutions. Assume p is of the form 6n + 1. p = 6n + 1, (p  1)/4 = 3n/2, (p + 1)/2 = 3n + 1; 3n/2 can be prime only when n = 2, therefore p = 13. 
October 9th, 2010, 02:19 AM  #5  
Global Moderator Joined: Dec 2006 Posts: 18,444 Thanks: 1462  Quote:
 
October 9th, 2010, 06:54 AM  #6 
Math Team Joined: Apr 2010 Posts: 2,778 Thanks: 361  Re: 2 elementary number theory problems....
Thanks, CRGreathouse, I'll use . I unfortunately can't edit anymore, so i'll replace a part: So we have: And than factor out mod 2 is in the exponent. Would that help? Or is there a "better" alternative? Hoempa 

Tags 
elementary, number, problems, theory 
Thread Tools  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
Book on elementary number theory  greg1313  Math Books  15  April 29th, 2015 02:36 AM 
What to include on a first elementary number theory course?  matqkks  Number Theory  2  June 17th, 2013 11:40 AM 
NUMBER theory PROBLEMS  MATHS FRIEND  Math Events  1  July 26th, 2012 02:25 PM 
elementary number theory without zero or negative numbers 2  tinynerdi  Number Theory  0  November 29th, 2009 11:01 PM 
elementary number theory without zero or negative numbers  tinynerdi  Number Theory  0  November 29th, 2009 10:25 PM 