Results are out! Find what you need...fast. Get quick advice or join the chat
Hey there Sign in to join this conversationNew here? Join for free

proof for equation with no solution

Announcements Posted on
    • Thread Starter
    • 0 followers
    Offline

    ReputationRep:
    How to prove that there is no solution for this equation in natural numbers:
    • 29 followers
    Online

    Consider modulo arithmetic.

    I could tell you what base works but not the justification for choosing it, as my number theory is too rusty.

    Is this really "Secondary school" level?
    • 2 followers
    Offline

    ReputationRep:
    Could you not use the quadratic formula?

     3n^2 + 3n + 7 = k^3

3n^2 + 3n + 7 - k^3 = 0

     \dfrac{-b\pm \sqrt{b^2 - 4ac}}{2a}

= \dfrac{ -3 \pm \sqrt{9 - 12(7-k^3)}}{6} ...

    I then wouldn't really know where to go from there, sorry.
    I just remember seeing something like this a couple of ears ago.
    • Thread Starter
    • 0 followers
    Offline

    ReputationRep:
    But quadratic formula is not proving anything i think ,i know that we can prove it by modulo but how ?
    • 29 followers
    Online

    (Original post by MAA_96)
    i know that we can prove it by modulo but how ?
    With the right base, the LHS can only be equal to certain values, and the RHS can only be equal to certain other values; and those two sets of values do not intersect, and hence there is no solution.

    Base is

    Spoiler:
    Show

    9
    • 3 followers
    Offline

    ReputationRep:
    3n^2+3n+7 =1 mod 3. Thus K^3 must be 1 mod 3. let K=3m+1 imples
    3n^2+3n+7=27(m^3+m^2)+9m+1
    n^2+n+2=9(m^3+m^2)+3m
    rhs=0 mod 3. n^2+n+2 is either 2 or 1 mod 3, never 3 by considering residue classes
    done
    • 11 followers
    Offline

    ReputationRep:
    (Original post by Blutooth)
    3n^2+3n+7 =1 mod 3. Thus K^3 must be 1 mod 3. let K=3m+1 imples
    3n^2+3n+7=27(m^3+m^2)+9m+1
    n^2+n+2=9(m^3+m^2)+3m
    rhs=0 mod 3. n^2+n+2 is either 2 or 1 mod 3, never 3 by considering residue classes
    done
    Clever. I wasn't thinking.
    • 3 followers
    Offline

    ReputationRep:
    (Original post by wcp100)
    Clever. I wasn't thinking.
    Thank you, but if you want clever I just solved an IMO problem in the summer maths thread. Took me 2 days to work out and I was thinking
    • 29 followers
    Online

    (Original post by Blutooth)
    ...
    Nice; for some reason I was getting 2 when thinking of 7 (mod 3) ; brain dead.

    Defo. not "secondary school".

    +rep.

Reply

Submit reply

Register

Thanks for posting! You just need to create an account in order to submit the post
  1. this can't be left blank
    that username has been taken, please choose another Forgotten your password?
  2. this can't be left blank
    this email is already registered. Forgotten your password?
  3. this can't be left blank

    6 characters or longer with both numbers and letters is safer

  4. this can't be left empty
    your full birthday is required
  1. By joining you agree to our Ts and Cs, privacy policy and site rules

  2. Slide to join now Processing…

Updated: July 6, 2012
New on TSR

GCSE results day

Waiting for your grades? Let off some steam in our results chat megathread

Article updates
Reputation gems:
You get these gems as you gain rep from other members for making good contributions and giving helpful advice.