Show that for all integers n>2, n does not divide n^2+2

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Homework Help Overview

The problem involves showing that for all integers n greater than 2, n does not divide n^2 + 2. The subject area pertains to number theory and divisibility.

Discussion Character

  • Exploratory, Assumption checking, Mathematical reasoning

Approaches and Questions Raised

  • One participant suggests using mathematical induction but expresses uncertainty about the recursive phrasing. Another participant questions the appropriateness of induction and proposes considering the divisibility of n^2. A further post explores the implications of assuming n divides n^2 + 2 and begins to manipulate that assumption.

Discussion Status

The discussion is active, with participants exploring different approaches to the problem. Some guidance has been offered regarding the use of induction and the properties of divisibility, but there is no explicit consensus on the best method to proceed.

Contextual Notes

Participants are working under the constraint that n must be greater than 2, and there is a focus on understanding the implications of divisibility in the context of the problem.

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Homework Statement


Show that for all integers n>2, n does not divide n^2+2.

2. The attempt at a solution
I believe this solution can be solved by induction, I just don't know how to phrase it recursively.

For all n>2, n^2+2 mod n ≠ 0

Base case n=3
3^2 + 2 =11
11 mod 3 = 2 ≠ 0.

Show that
(n-1)^2 +2 mod n-1 ≠ 0
 
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I don't think induction is the right way to do this. n divides n2, can you use that in this problem?
 
I can use that, but I'm not sure how it would be done. Intuitively it makes sense, but is there an axiom that I could use that proves this?
 
Suppose n does divide n2+ 2. Then there exist integer m such that mn= n2+ 2. Now divide both sides by n.
 

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