Are positive integers of the form $p^2+p+1$ known to be special?

Solution 1:

The expression $n^2+n+1$ is a special case $(z,w)=(n,1)$ of $z^2+zw+w^2$ which definitely is well-studied.

Define a function $r:\mathbb C^2\to\mathbb C$ by $$r(z,w)=r_1z^2+r_2zw+r_3w^2$$ such that $r_1,r_2,r_3\in\mathbb C$. This is called a binary form of degree 2.

For $a,b,c,d\in\mathbb C$, given the condition that $ad-bc=1$, we can substitute: $$z\mapsto az+cw$$ $$w\mapsto bz+dw$$

And this will yield another polynomial that has the same (or an invariant) form: $$r'(z,w)=r(az+cw,bz+dw)=r_1'z^2+r_2'zw+r_3'w^2$$

In other words: let $S=\begin{pmatrix} a & b \\ c & d \end{pmatrix}$ such that $\det S=1$. Left-multiplying $S$ by $(z\;\;\; w)$ yields: $$(z\;\;\; w)\times \begin{pmatrix} a & b \\ c & d \end{pmatrix} = (az+cw\;\;\; bz+dw)$$ and so the substitution $(z,w)\mapsto (az+cw, bz+dw)$ is just a linear combination of the rows of $S$ with coefficients from the row vector $(z\;\;\; w)$.

Now a question is: are the coefficients $(r_1,r_2,r_3)$ and $(r_1',r_2',r_3')$ also related by matrix multiplication? The answer is: yes (but with column vectors)! In fact:

$$\begin{pmatrix} r_1' \\ r_2' \\ r_3' \end{pmatrix} = \begin{pmatrix} a^2 &ab &b^2 \\ ac &ad+bc &bd \\ c^2 &cd &d^2 \end{pmatrix} \times \begin{pmatrix} r_1 \\ r_2 \\ r_3 \end{pmatrix}$$ And the coefficients elegantly satisfy a simple identity: $$r_2^2-4r_1r_3=(r_2')^2-4r_1'r_3'$$

So more generally, the expression $n^2+n+1$ can be given by $r(z,w)$ where $z=n$ and $w=1=r_1=r_2=r_3$. This whole concept (called classical invariant theory) is not unique to prime arguments though, but what is studied about the expression does contain this concept and I think it is still pretty special.

Solution 2:

The number $n^2+n$ is a pronic number (also called an oblong or rectangular number); equivalently, these are twice a triangular number $T_n = n(n+1)/2$.

Fermat studied these numbers a great deal, included the special case you highlight where $n^2+n+1= 2T_n+1 = p$ is prime. In that circumstance, the composition of $T_n$ is connected with numbers which are the sum of two squares (something Fermat spent a great deal of time studying).