February 9th, 2010, 05:23 PM  #1 
Newbie Joined: Sep 2009 Posts: 11 Thanks: 0  x^2 congruent to 2 mod 5
Hi! I need some help on this problem: How to prove has no solutions? (x is an integer) Any help is appreciated! 
February 9th, 2010, 05:50 PM  #2 
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Re: x^2 congruent to 2 mod 5
Compute the squares mod 5. 4^2 = 1, 3^2 = 4, etc.

February 9th, 2010, 06:52 PM  #3 
Newbie Joined: Sep 2009 Posts: 11 Thanks: 0  Re: x^2 congruent to 2 mod 5
Hello, CRGreathouse! Thank you for your advice. Since all nonnegative integers can be expressed as: 5k, 5k + 1, 5k + 2, 5k + 3, 5k + 4 (k can be any nonnegative integer), I found out something useful (I think): and have proven it. Thus the sequence looks like 0, 1, 4, 4, 1, 0, 1, 4, 4, 1, 0... and there is no 2. But are there any more convenient ways to solve this problem? Thank you again! 
February 9th, 2010, 09:01 PM  #4  
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Re: x^2 congruent to 2 mod 5 Quote:
 
February 10th, 2010, 11:16 PM  #5  
Newbie Joined: Sep 2009 Posts: 11 Thanks: 0  Re: x^2 congruent to 2 mod 5 Quote:
And are there general ways to determine whether congruences like x^2 = a (mod b) have solutions? Thanks!  

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