Prove Prime p^2+2 is Composite mod 3 #5

  • Context: Graduate 
  • Thread starter Thread starter phyguy321
  • Start date Start date
  • Tags Tags
    Proof
Click For Summary

Discussion Overview

The discussion revolves around proving that for prime numbers \( p \geq 5 \), the expression \( p^2 + 2 \) is composite, with a focus on modular arithmetic, particularly modulo 3. Participants are exploring hints and previous related posts to aid in the proof.

Discussion Character

  • Homework-related
  • Mathematical reasoning

Main Points Raised

  • One participant suggests working modulo 3 and using a previous post (#5) to demonstrate that \( p^2 + 2 \) has a factor of 3.
  • Another participant expresses confusion about modular arithmetic and requests guidance on how to approach the problem.
  • A participant references a previous post that discusses the properties of integers modulo 3, specifically that \( n^2 \) is either 0 or 1 modulo 3.
  • There is a suggestion to analyze the expression \( n^2 + 2 \) in the context of modulo 3.

Areas of Agreement / Disagreement

Participants do not appear to reach a consensus on how to start the proof, with some expressing confusion about modular arithmetic and others attempting to guide them. The discussion remains unresolved as participants seek clarification and assistance.

Contextual Notes

There is a lack of clarity regarding the definitions and implications of modular arithmetic in this context, as well as the specific content of post #5, which may be crucial for understanding the problem.

phyguy321
Messages
45
Reaction score
0
Prove that if p [tex]\geq[/tex] 5 is prime, then p[tex]^{2}[/tex] +2 is composite
(hint: work mod 3 and use #5 to show p^2 + 2 has a factor of 3.)
 
Physics news on Phys.org
phyguy321 said:
Prove that if p [tex]\geq[/tex] 5 is prime, then p[tex]^{2}[/tex] +2 is composite
(hint: work mod 3 and use #5 to show p^2 + 2 has a factor of 3.)

Hi phyguy321! :smile:

Show us what you've tried, and where you're stuck, and then we'll know how to help. :smile:

(and … erm … what's #5? :redface:)
 
The problem is i have no idea of where to start. Modular arithmetic makes no sense to me. and #5 was my other post on modular arithmetic
 
phyguy321 said:
The problem is i have no idea of where to start. Modular arithmetic makes no sense to me. and #5 was https://www.physicsforums.com/showthread.php?t=261171"

Your other post was to prove that for any integer n, n2 = 0 or 1 (mod 3).

So n2 + 2 = … ? (mod 3).
 
Last edited by a moderator:

Similar threads

Undergrad How to Prove a Congruence Relation for Prime Numbers?
  • · Replies 17 ·
Replies
17
Views
2K
Undergrad Trouble understanding an online solution to an exercise in Dummit & Foote
  • · Replies 21 ·
Replies
21
Views
2K
Undergrad Fundamental theorem of arithmetic from Jordan-Hölder theorem
  • · Replies 5 ·
Replies
5
Views
3K
Undergrad How to show ##p(x)=g(x)x\pm 1\in\Bbb{Q}[x]## is irreducible in ##\Bbb{Q}_{\Bbb{Z}}[x]##?
  • · Replies 48 ·
2
Replies
48
Views
6K
Undergrad ##(A/\mathfrak{a})_{\mathfrak{p}/\mathfrak{a}}## and its isomorphism?
  • · Replies 31 ·
2
Replies
31
Views
3K
If p##\geq##5 is a prime number, show that p^{2}+2 is composite?
  • · Replies 3 ·
Replies
3
Views
2K
Undergrad Frequency of prime number gaps according to (p-1)/(p-2)
  • · Replies 2 ·
Replies
2
Views
2K
If ## p\geq q\geq 5 ## and ## p ## and ## q ## are both primes ....
  • · Replies 30 ·
2
Replies
30
Views
3K
MHB Proving $x-1$ is a Factor of $P(x)$ with Polynomials
  • · Replies 1 ·
Replies
1
Views
1K
If ## p ## and ## p^{2}+8 ## are both prime numbers, prove that....
  • · Replies 16 ·
Replies
16
Views
3K