Number Theory (Modular Arithmetic and Perfect Squares)

1. Oct 13, 2012

m166780

1. The problem statement, all variables and given/known data

If k is an integer, explain why 5k +2 cannot be a perfect square.

2. Relevant equations

n/a

3. The attempt at a solution
I'm in way over my head and not really sure what type of proof I should be using. In my course, we just went over some number theory and modular algebra so I'm pretty sure that this has something to do with this.
I've been researching this and the closest that I have found to similar problems are:

Prove that 3a2− 1 is never a perfect square.
Observe that 3a2− 1 = 3
(a^2− 1) + 2 = 3k + 2, for k = a2− 1.
The results of problem 3.a tell us that the square of an integer must either be of the
form 3k or 3k + 1. Hence, 3a2− 1 = 3k + 2 cannot be a perfect square.
http://www.pat-rossi.com/MTH4436/homework/hw_2_1_and_2_2.pdf [Broken]

These might be relevant also:
example 10
http://palmer.wellesley.edu/~ivolic/pdf/Classes/OldClassMaterials/MATH223NumberTheorySpring07/Homework4Solutions.pdf [Broken]

The "text" for this course are just handouts from the professor. The chapter in Mathematics: A Discrete Introduction might help for a reference if anyone has it. I can upload the notes too if those might help.

Last edited by a moderator: May 6, 2017
2. Oct 13, 2012

jbunniii

Every integer is of one of the following forms:

5m
5m+1
5m+2
5m+3
5m+4

Try squaring each of these forms and see if the result can be of the form 5k+2.