
Number Theory Number Theory Math Forum 
 LinkBack  Thread Tools  Display Modes 
September 16th, 2017, 08:05 PM  #1 
Newbie Joined: Sep 2017 From: San Diego Posts: 8 Thanks: 0  If P>= 5 is a prime number, then p^2 +2 is composite.
I have seen outlines of the proof, and I tried to fill in the reasons why. Are they right? If p>=5 is a prime number, then p^2 +2 is composite. Proof: Suppose p>=5 is a prime number. Then by the quotient remainder theorem, p can be expressed as 6m or 6m + 1 or 6m + 3 or 6m + 4 or 6m + 5 for some integer m. However, since p is a prime number greater than or equal to 5, it cannot be expressed as a multiple of 2 or 3 greater than 5, because p would then be not prime. Thus p can only be expressed as 6m+1 or 6m+5. If p = 6m + 1, then by squaring p and adding 2, we get p^2 + 2 = (6m+1)^2 + 2 = 3(12m^2 + 4m +1) Thus p^2 + 2 is divisible by a number less than p^2 + 2 and greater than 1, so, p^2 + 2 is composite. If p = 6m + 5, then by similar reasoning we get p^2 + 2 = (6m+5)^2 + 2 = 3(12m^2 + 20m +7) Thus p^2 + 2 is divisible by a number less than p^2 + 2 and greater than 1, so, p^2 + 2 is composite. Hence, in either case p^2 + 2 is composite, which is what we needed to show. 
September 16th, 2017, 11:36 PM  #2 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,091 Thanks: 2360 Math Focus: Mainly analysis and algebra 
Yes: $p = 6m \pm 1$ and so $p \equiv \pm 1 \pmod{6}$ and $p^2 \equiv 1 \pmod{6}$ and $p^2 + 2\equiv 3 \pmod{6}$ and thus $p^2+2 = 6k+3$.
Last edited by v8archie; September 16th, 2017 at 11:39 PM. 

Tags 
composite, number, p>, p>, prime 
Thread Tools  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
Composite numbers primelike  mobel  Number Theory  24  October 30th, 2014 10:18 AM 
Composite number  agustin975  Number Theory  8  March 8th, 2013 10:04 AM 
probability that a number is composite  ershi  Number Theory  113  August 18th, 2012 05:45 AM 
Integers, Prime, and Composite Numbers  Mighty Mouse Jr  Algebra  6  May 11th, 2010 10:08 AM 
Serial position of a composite number  Geir  Number Theory  1  July 11th, 2009 05:57 AM 