1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Number Theory (Modular Arithmetic and Perfect Squares)

  1. Oct 13, 2012 #1
    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


    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. jcsd
  3. Oct 13, 2012 #2


    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    Every integer is of one of the following forms:


    Try squaring each of these forms and see if the result can be of the form 5k+2.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook