Results 1 to 2 of 2

Math Help - Odd squares

  1. #1
    Senior Member I-Think's Avatar
    Joined
    Apr 2009
    Posts
    288

    Odd squares

    Good night.
    I have a proof question here that I needs validating. I was also wondering if I could receive alternative solutions for the question.
    Thanks in advance for the help

    Question
    Prove that the square of an odd integer is always of the form 8q+1 where q is an integer

    Answer
    We wish to prove that 8q+1=k^2, where k is an odd integer
    Let k be an odd integer of the form: 2I+1
    (2I+1)^2=4I^2+4I+1

    As I is an integer, it can either be odd or even
    Case 1: I is odd, i.e. it is of the form: 2m+1
    (2I+1)^2= 4(2m+1)^2+4(2m+1)+1
    16m^2+16m+4+8m+4+1=16m^2+24m+9
    (16m^2+24m+8)+1
    8(2m^2+3m+1)+1 Let 2m^2+3m+1=q
    8q+1

    Case 2: I is even, i.e. it is of the form 2m
    (2I+1)^2= 4(2m)^2+4(2m)+1
    16m^2+8m+1=(16m^2+8m)+1
    8(2m^2+m)+1 Let 2m^2+m=q
    8q+1

    Proven in both cases, hence the statement is true.
    End of solution. Q.E.D

    I welcome your critiques and say thanks in advance,
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member

    Joined
    May 2006
    From
    Lexington, MA (USA)
    Posts
    11,719
    Thanks
    635
    Hello, I-Think!

    Your proof is correct.


    Prove that the square of an odd integer is always of the form 8q+1 where q is an integer.

    Let the odd integer be: . n \:=\:2p+1\,\text{ for some integer }p.

    . . Then: . n^2 \:=\:(2p+1)^2 \:=\:4p^2 + 4p + 1


    Here is a very sneaky step . . .

    \text{We have: }\;n^2 \;=\;4\underbrace{p(p+1)} + 1\;\;{\color{blue}[1]}
    . . . . . . . .
    two consecutive integers

    With two consecutive integers, one of them is even.
    . . Hence, their product is even: . p(p+1) \:=\:2q\,\text{ for some integer }q.

    Then {\color{blue}[1]} becomes: . n^2 \;=\;4(2q) + 1 \;=\;8q+1 \quad\hdots\;\text{ta-}DAA!

    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. sum of squares
    Posted in the Number Theory Forum
    Replies: 1
    Last Post: October 16th 2010, 06:25 PM
  2. Magic squares of Squares
    Posted in the Math Puzzles Forum
    Replies: 5
    Last Post: September 22nd 2010, 09:58 AM
  3. Squares
    Posted in the Math Puzzles Forum
    Replies: 6
    Last Post: December 2nd 2009, 06:55 PM
  4. Replies: 4
    Last Post: November 13th 2009, 05:12 PM
  5. sum of squares
    Posted in the Algebra Forum
    Replies: 1
    Last Post: September 7th 2007, 07:19 AM

Search Tags


/mathhelpforum @mathhelpforum