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If $n$ is a positive integer with $n\equiv 2\pmod 3$ then I want to show that each odd divisor of $n^2+n+1$ is congruent to $1\pmod 3$.

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I have done the following:

Let $d$ be an odd divisor of $n^2+n+1$.

Then $$n^2+n+1\equiv 0\bmod d \Rightarrow (n-1)(n^2+n+1)\equiv 0\bmod d \Rightarrow n^3-1\equiv 0 \bmod d \Rightarrow n^3\equiv 1\bmod d$$

Since $n\equiv 2\pmod 3$ we have that $n=2+3k$. Then $$n^3=(2+3k)^3=27k^3+54k^2+36k+8$$

We have that $n^3\equiv 1\bmod d$ therefore we get $$27k^3+54k^2+36k+8\equiv 1\bmod d \Rightarrow 27k^3+54k^2+36k+7\equiv 0\bmod d$$

Is everything correct so far? How could we continue?

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    $\begingroup$ Try for prime divisors and consider the multiplicative order of $n$ mod $p$. Note that $n^2+n+1$ is odd. $\endgroup$ – Mindlack Dec 15 '18 at 20:57
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Hint: Start with odd prime divisors. As you have correctly shown, for any $p\mid n^2+n+1$ we have that $$n^3\equiv 1\pmod p.$$ Hence $\text{ord}_{p}(n)\mid 3$. Since $n\equiv 2\not\equiv 1\pmod p$ we know that $\text{ord}_{p}(n)=3$. However, we also know that for every $m$ and $x$ coprime to $m$ that $\text{ord}_m(x)\mid \phi(m)$. How can we apply this in this scenario? Once we know the statement holds for every odd prime divisor, what can we say about every odd divisor?

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  • $\begingroup$ So, in this case we have that $\text{ord}_p(n)\mid \phi(p)\Rightarrow 3\mid p-1 \Rightarrow p-1=3k \Rightarrow p=3k+1$. That means that $p\equiv 1\pmod 3$. Any odd divisor will be a product of odd prime divisors, right? So we have to show that the product of terms in the form $3m+1$ will be again in that form, right? $\endgroup$ – Mary Star Dec 15 '18 at 21:23
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    $\begingroup$ @MaryStar Exactly, but what is a product of things $\equiv 1\pmod 3$ congruent to mod 3? $\endgroup$ – Will Fisher Dec 15 '18 at 21:24
  • $\begingroup$ It is congruent to $1$ modulo $3$, since let $a\equiv b\equiv 1\bmod 3$ then $ab\bmod 3\equiv (a\bmod 3 )(b\bmod 3)\bmod 3\equiv 1\bmod 3$, right? $\endgroup$ – Mary Star Dec 15 '18 at 21:51
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    $\begingroup$ @MaryStar Yep. And like you said, odd divisors are products of odd primes, all of which we know are congruent to 1 mod 3. $\endgroup$ – Will Fisher Dec 15 '18 at 21:53
  • $\begingroup$ Great!! Thanks a lot!! :-) $\endgroup$ – Mary Star Dec 15 '18 at 21:59

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