June 7th, 2017, 06:32 AM  #1 
Math Team Joined: Jul 2011 From: North America, 42nd parallel Posts: 3,372 Thanks: 233  Elementary Number Theory question
Show that $ \ \ 7 \ \ $ does not divide $ \ \ n^2 + 1 \ \ $ for any integer $ \ \ n \ \ $ 
June 8th, 2017, 04:40 AM  #2 
Math Team Joined: Jul 2011 From: North America, 42nd parallel Posts: 3,372 Thanks: 233 
Hint: A possible approach is to use the theory of congruences Of interest may be the following sequence in the Online Encyclopedia Of Integer Sequences https://oeis.org/A192450 
June 8th, 2017, 05:52 AM  #3  
Senior Member Joined: Feb 2010 Posts: 663 Thanks: 119  Quote:
In mod 7, $\displaystyle n$ must be in $\displaystyle \{0,1,2,3,4,5,6\}$ So $\displaystyle n^2$ must be in $\displaystyle \{0,1,2,4\}$ Thus, $\displaystyle n^2+1$ must be in $\displaystyle \{1,2,3,5\}$ Therefore, $\displaystyle n^2+1$ is never 0 mod 7.  
June 8th, 2017, 07:22 AM  #4 
Math Team Joined: Jul 2011 From: North America, 42nd parallel Posts: 3,372 Thanks: 233 
Yes, very good. A clear and precise explanation. Did you look at the oeis link? It looks like 83 out of the first 108 natural numbers do not divide $ \ \ n^2 + 1 \ \ $ which is surprising to me. What are your thoughts about that? Last edited by skipjack; June 9th, 2017 at 01:31 AM. 
June 9th, 2017, 01:54 PM  #5  
Senior Member Joined: Nov 2010 From: Berkeley, CA Posts: 174 Thanks: 35 Math Focus: Elementary Number Theory, Algebraic NT, Analytic NT  Quote:
The condition on n is equivalent to the congruence $x^2 \equiv 1 \pmod{n}$ not having a solution for $x$ (We also say that 1 is a quadratic nonresidue mod n.). Write n in its canonical prime decomposition: $n=2^{a_1}3^{a_2} \cdots p_k^{a_k}$ For the purpose of this explanation, disregard 2 if it divides n. As is often the case, 2 has to be handled separately. Then it's known that the preceding congruence has a solution if and only if each of the congruences $x^2 \equiv 1 \pmod{p_i}$ has a solution. It's known that $x^2 \equiv 1 \pmod{p}$ (where p is an odd prime) has a solution if $p \equiv 1 \pmod{4}$ and does not have a solution if $p \equiv 3 \pmod{4}$. We conclude that the original congruence does not have a solution if just one of the ${p_i}\equiv 3 \pmod{4}$. Conversely, the original congruence has a solution only if all the ${p_i} \equiv 1 \pmod{4}$. Evidently, there are more of the former than the latter.  
June 10th, 2017, 04:03 AM  #6 
Math Team Joined: Jul 2011 From: North America, 42nd parallel Posts: 3,372 Thanks: 233 
Thank you for your nice reply. I did not know that there must be a solution for all $ \ \ p_i \ \ $ of the prime factorization of the divisor. Very interesting ... so if the divisor contains at least one prime of the form $ \ \ 4r + 3 \ \ $ then it will not divide $ \ \ n^2 + 1 \ \ $ On the flipside ... a natural number like $ \ \ 2 \times 5 \times 13 = 130 \ \ $ is sure to divide $ \ \ n^2 + 1 \ \ $ because all the prime factors of $ 130 \ \ $ divide $ \ \ n^2 + 1 \ \ $ Is that right? Please feel free to post any other details you see fit. Last edited by agentredlum; June 10th, 2017 at 04:18 AM. 
June 11th, 2017, 09:49 AM  #7 
Senior Member Joined: Nov 2010 From: Berkeley, CA Posts: 174 Thanks: 35 Math Focus: Elementary Number Theory, Algebraic NT, Analytic NT 
Yes, your statements are correct. For more details on moving back and forth between congruences whose moduli are a single prime, to moduli that are multiple powers of a prime and to the general case, see this link.

June 11th, 2017, 01:21 PM  #8 
Math Team Joined: Jul 2011 From: North America, 42nd parallel Posts: 3,372 Thanks: 233 
The link does not work in my browser

June 11th, 2017, 03:17 PM  #9 
Senior Member Joined: Nov 2010 From: Berkeley, CA Posts: 174 Thanks: 35 Math Focus: Elementary Number Theory, Algebraic NT, Analytic NT 
Does this link work? https://www.johndcook.com/blog/quadratic_congruences/ If not, what error message, if any, do you get? 
June 12th, 2017, 06:55 AM  #10  
Math Team Joined: Jul 2011 From: North America, 42nd parallel Posts: 3,372 Thanks: 233  Quote:
I do not own a working computer, using a PS3. The error is probably resulting from the PS3 browser's inability to decode Adobe but this is speculation on my part. This has been going on for many years , 10 years at least and I'm not the only one with this problem. The error is ... The page cannot be displayed. (80710a06) ty for your concern in this matter  

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