6
$\begingroup$

The problem: Prove that there exist no positive integers $m$ and $n$ for which $m^2+m+1=n^2$.

This is part of an introductory course to proofs, so at this point, the mathematical machinery should not be too involved. This is supposed to be proven by contradiction. I've been messing around with this for a bit and can't help but feel that I'm missing something completely obvious. My first instinct was to evaluate this case by case based off of combinations of even and odd for m and n, which led to contradictions in the case that both m and n are even or that m is odd and n is even (the contradiction being that zero equates to an odd number). The problem comes when trying to find a contradiction where n is a positive odd number and m is either even or odd.

I then tried to approach it by showing that if $n^2=m^2+m+1$ and n is a positive integer, that $(m^2+m+1)^{\frac{1}{2}}$ must be a positive integer, and that this leads to a contradiction. It certainly looks like this will be the case, since, for the first few positive integer values of m, we get $3^{\frac{1}{2}}$,$7^{\frac{1}{2}}$,$13^{\frac{1}{2}}$,$21^{\frac{1}{2}}$, none of which are positive integers, but at this point at least, I don't know how to demonstrate this with a proof.

A gentle nod in the correct direction would be greatly appreciated. Just starting off with this stuff, so any help/insight would be great.

$\endgroup$
  • 4
    $\begingroup$ $m^2 < m^2 + m + 1 < m^2 + 2m + 1 = (m+1)^2$. $\endgroup$ – Daniel Fischer Oct 5 '13 at 13:06
15
$\begingroup$

Hint: If such $n$ existed, then it would have to be between $m$ and $m+1$ since $$m^2<m^2+m+1<m^2+2m+1=(m+1)^2.$$

$\endgroup$
  • 1
    $\begingroup$ This is an answer, and not just a hint. :-) I've replaced "such number" with "$n$", to make it a bit more clear. $\endgroup$ – Vedran Šego Oct 5 '13 at 13:12
  • $\begingroup$ @VedranŠego: Thanks. It's hard to provide a so-called hint to this question, but it might still require some thought to convince one's self why the implication is true. I'm hoping, anyway, since I called it a hint. :) $\endgroup$ – Clayton Oct 5 '13 at 13:14
  • $\begingroup$ I see it now. So it's m^2<m^2+m+1=n^2<(m+1)^2, and since m and m+1 are successive positive integers, we must have m<n<m+1, which is impossible. Thanks very much for the help. It is greatly appreciated. $\endgroup$ – mrmingus Oct 5 '13 at 13:37
5
$\begingroup$

Clayton's answer is certainly simpler that this, but here's an alternative.

Multiply both sides by $4$ and you get:

$$(2n)^2 = 4m^2+4m+4 = (2m+1)^2 + 3$$

So $$(2n-(2m+1))(2n+(2m+1)) = 3$$

This means that $3$ can be factored into two numbers that differ by $4m+2$. That means $m=0$ or $m=-1$.

$\endgroup$
0
$\begingroup$

I am not certain that this counts as a proof, but here it is: Rewrite the formula: $m^2+m=n^2-1$ gives $m(m+1)=(n+1)(n-1)$. Left side factors are consecutive integers, one odd and one even but right side factors are not consecutive, both are odd or both are even.The smaller factors should equal each other and so should the bigger factors: $m=n-1$ and $m+1=n+1$ which contradicts each other. So there does not exist positive integers m and n for which $m^2+m+1=n^2$.

$\endgroup$
  • $\begingroup$ Point 1: Both factors $(n+1)(n-1)$ can't be odd, since one of $m$ or $m+1$ must be even, making the product $m(m+1)$ even. Therefore both factors $n-1$ and $n+1$ must be even. $\endgroup$ – James Arathoon Feb 1 at 20:42
  • $\begingroup$ Point 2: You haven't justified the statement "The smaller factors should equal each other and so should the bigger factors" Since $n-1$ and $n+1$ must both be even, take a factor of four from either $m$ or $m-1$ and write $2a=n-1$ and $2b=n+1$. Now the smallest difference between $a$ and $b$ is when for example $b=a+1$ or $b=a-1$. Can you conclude anything from this? $\endgroup$ – James Arathoon Feb 1 at 20:55
  • $\begingroup$ Yes, I can. Thank you! $\endgroup$ – Ylvali Feb 3 at 12:54

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.