# Are all primes the same?

Are there properties "intrinsic"* to primes that set them apart?

Example 1: The fields $\mathbb{F}_p$ do not all have the same structure. If $p=1 \mod 4$ is a prime then there is a number $1<x<p$ where $x^{-1}=-x\mod p$, i.e. in the field $\mathbb{F}_p$, its multiplicative inverse is also its additive inverse. Moreover, this happens always and only when $p=1\mod 4$ and not when $p=3\mod 4$. Thus the odd primes are neatly divided into two classes with distinctly different properties.

Example 2: The fields $\mathbb{F}_p$ yield good expander graphs, but some are better than others. For a prime $p$, construct an undirected graph on the vertices $\{0,1,\ldots,p-1\}$ as follows: node $n$ is connected to $n-1,n+1$ and $n^{-1}\mod p$. Let $\lambda_p$ be the spectral gap of the graph thus obtained (the difference between the two largest eigenvalues of the adjacency matrix). It follows from Selberg's 3/16 theorem that the resulting graph is a Ramanujan graph: it has an almost maximal spectral gap, hence these graphs are good expanders, and their diameter will be very small: only $\mathcal{O}(\log p)$. But some primes yield better expanders than others. I'm looking at a plot for the spectral gaps of primes up to 727, and there are no discernable local trends, although it looks like the spectral gap might converge to the expected value of $3-\sqrt{8}$.

Are there more examples that show that not all primes are equal? This question is born of curiosity, so feel free to bring surprising results from any area of mathematics!

*Of course there are twin primes, Sophie Germain primes and Mersenne primes, but these labels only show where a prime is located with respect to other numbers. For the purposes of this question, I am interested in properties "intrinsic" to a prime that set it apart from other primes.

• you want to find other "neat" ways to classify the primes into two groups? – Jorge Fernández Hidalgo Mar 22 '18 at 15:23
• The first "neat" property in your question is $-1$ is a quadratic residue mod $p$. – robjohn Mar 22 '18 at 15:34
• @JorgeFernández Yes! Or, the second example associates a real number with every prime. Anything that distinguishes between primes is fair game, except by comparing primes to other neighbouring numbers, like twin primes ("$p$ and $p+2$ are both prime) or Sophie Germain primes ($2p+1$ is also prime). What is it about those primes that makes them interesting? – Lieuwe Vinkhuijzen Mar 22 '18 at 15:44
• Do also properties count that single out just one prime? For example, only for $p=2$ there exists $0<n<p$ such that $n\equiv -n \pmod p$. – celtschk Mar 22 '18 at 15:44
• @robjohn That's correct! I phrased it this way to emphasize that the field $\mathbb{F}_p$ has a special property, namely sometimes multiplicative and additive inverses coincide. Phrased as a quadratic residue, this property may seem somewhat arbitrary, for why not ask: is $42$ a quadratic residue? – Lieuwe Vinkhuijzen Mar 23 '18 at 15:49

There are plenty of obvious elementary properties separating primes of the form $4n+1$ and $4n+3$, e.g., quadratic residue's $$(-1/p)=(-1)^{\frac{p-1}{2}},$$ or the product $(4n+1)(4m+1)$ is closed, i.e., again of the form $4k+1$, but not for $4n+3$. There are also deeper properties, for example Chebyshev's bias. There are "more primes of the form $4n + 3$ than of the form $4n + 1$". This has become popular under the name Prime number races. This is related to GRH, the Generalized Riemann Hypothesis. For a reference, see for example here.
You can divide the primes into two types: those primes $p$ such that $p+1$ has exactly one prime factor and the primes $p$ such that $p+1$ has more than one prime factor. Of course, the primes of the first type are exactly the Mersenne primes.