prime $p$ different from $3$ is of the form $3k+1$ if and only if $p$ divides number of form $c^2+c+1$

Question

Prove that prime $p$ different from $3$ is of the form $3k+1$ if and only if $p$ divides number of from $c^2+c+1$

i am trying but i did't get answer any hint please

Answer 1

Notice that if $p\equiv 1$ mod 3, then $3$ divides $p-1=\phi(p)$ where $\phi$ is the euler function. This implies that there exists $c$ not congruent to 1 mod $p$ such that $c^3\equiv 1$ mod p, (this is a fact from group theory). From this you can conclude that $p$ divides $c^3-1=(c-1)(c^2+c+1)$. Since $p$ doesn't divide $c-1$ you have that $c^2+c+1$.

The other direction shouldn't be that hard.

(I just noticed that this is the solution Jyrki said in the comments).

Answer 2

Solution with Legendre symbol.

Say $p=6k+r$ where $r\in \{1,5\}$ ($r$ can't be $0,2,3$ and $4$).

\begin{eqnarray*} p\mid c^2+c+1 &\Longleftrightarrow & p\mid (2c+1)^2+3\\ &\Longleftrightarrow & -3\equiv_p (2c+1)^2\\ &\Longleftrightarrow & \Big({-3\over p}\Big) =1 \\ &\Longleftrightarrow & \Big({-1\over p}\Big)\cdot \Big({3\over p}\Big)=1\\ &\Longleftrightarrow & \Big({-1\over p}\Big)\cdot \Big({3\over p}\Big)=1\\ &\Longleftrightarrow & (-1)^{p-1\over 2}\cdot (-1)^{\big[{p+1\over 6} \big]}=1\\ &\Longleftrightarrow & {p-1\over 2}+ \Big[{p+1\over 6} \Big]\equiv_2 0\\ &\Longleftrightarrow & 3k+{r-1\over 2}+ k\Big[{r+1\over 6} \Big]\equiv_2 0\\ &\Longleftrightarrow & {r-1\over 2}+ \Big[{r+1\over 6} \Big]\equiv_2 0\\ &\Longleftrightarrow & r=1 \end{eqnarray*}

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Second solution (only one way)

Perhaps it would be easyer to see that $$c^2+c+1 \equiv _6 1,3$$ Each prime that divide $c^2+c+1$ is $3$ or $6k+r$ where $r=1$ or $r=5$. It is easy to see that the sets $$A=\{6a+1;\;a\in \mathbb{Z}\}$$ and $$B= \{6b+5;\;b\in \mathbb{Z}\}$$ are closed for multiplication. And if we multiply $x\in A$ and $y\in B$ we get $xy\in B$.

So if there is at least one $p \equiv_6 5$, then product of all primes would be $\equiv_6 5$ but this can be.

Answer 3

Solution using Einsenstein integers

The Eisenstein ring $E:=\mathbf Z [\omega]$ is the ring of integers of the quadratic field $\mathbf Q(\omega)$, where $\omega$ is a primitive cubic root of unity. $E$ is a PID (even a Euclidian domain) in which the decomposition of the rational primes $p$ is classically known : (i) $p=3$ is totally ramified, more precisely $3$ is associate to $(1-\omega)^2$ ; (ii) $p\equiv-1\pmod 3$ is inert, i.e. $p$ remains prime in $E$ ; (iii) $p\equiv 1\pmod 3$ is totally split, i.e. $p=\pi.\bar{\pi}$, the product of two conjugates primes of $E$. The splitting case is equivalent to $p=N(\pi)$ , where the norm $N$ is such that $N(a+b\omega)=a^2 + b^2 - ab$ (see the hint of @Oscar Lanzi). Since $E$ is a PID, for any prime $\pi$, the quotient $E/\pi E$ is a finite field of order $N(\pi)$, and case (iii) is equivalent to saying that $\mathbf Z/p\mathbf Z \cong E/\pi E$, or that $ (E/\pi E)^*$ is a cyclic group of order $N(\pi)-1=p-1$.

Now, sticking to case (iii), fix a generator $g$ of $(E/\pi E)^*$, so that $x=g^{\frac {p-1}{3}}$ is a cubic root of unity in $(E/\pi E)^*$. But $x^3 - 1= (x-1)(x-\omega)(x-\omega^2)$ in $ (E/\pi E)^*$, where the three cubic roots of unity are easily seen to be distinct. It follows that case (iii) is equivalent to saying that $g^{\frac {p-1}{3}}$ is a root of $(x-\omega)(x-\omega^2)=x^2 +x + 1$. Using the isomorphism $\mathbf Z/p\mathbf Z\cong E/\pi E$, this is in turn equivalent to $c^2+c+1=0$ in $(\mathbf Z/p\mathbf Z)^*$, where $c$ corresponds to $g^{\frac {p-1}{3}}$ .

Of course, the above argument appears unduly complicated compared to those of @Jose Cruz or @John Watson. But it presents the advantage to surrepticiously introduce Legendre's symbol of cubic residues, which can be defined as follows. In both cases (ii) and (iii), if $\pi$ is a prime of $E$ not dividing $\alpha \in E$, obviously $\alpha^{N(\pi)-1}\equiv 1\pmod \pi$. Note that $N(\pi)=p^2$ in case (ii), so that, as previously, $\alpha^{\frac{N(\pi)-1}{3}}\equiv 1, \omega,\omega^2 \pmod \pi$ in both cases (ii) and (iii). The Legendre symbol $(\alpha /\pi)_3$ is then defined to be the unique cubic root of unity s.t. $\alpha^{\frac{N(\pi)-1}{3}}\equiv (\alpha /\pi)_3\pmod \pi$. The cyclicity of $(E/\pi E)^*$ implies that $(\alpha /\pi)_3=1$ iff $x^3\equiv \alpha \pmod \pi$ has a solution in $E$ .