Reversible primes

Abstract: The properties of the digits of prime numbers and various other sequences of integers have attracted great interest in recent years. For any positive integer $k$, we denote by $\overleftarrow{k}$ the reverse of $k$ in base 2, defined by $\overleftarrow{k} = \sum_{j=0}^{n-1} \varepsilon_j\,2^{n-1-j}$ where $k = \sum_{j=0}^{n-1} \varepsilon_{j} \,2^j$ with $\varepsilon_j \in \{0,1\}$, $j\in\{0, \ldots, n-1\}$, $ \varepsilon_{n-1} = 1$. A natural question is to estimate the number of primes $p\in \left[2^{n-1},2^n\right)$ such that $\overleftarrow{p}$ is prime. We will present a result which provides an upper bound of the expected order of magnitude. Our method is based on a sieve argument and also allows us to obtain a strong lower bound for the number of integers $k$ such that $k$ and $\overleftarrow{k}$ have at most 8 prime factors (counted with multiplicity). We will also present an asymptotic formula for the number of integers $k\in \left[2^{n-1},2^n\right)$ such that $k$ and $\overleftarrow{k}$ are squarefree.

This is a joint work with Cécile Dartyge, Bruno Martin, Joël Rivat and Igor Shparlinski.

dynamical systemsnumber theory

Audience: researchers in the topic

( paper | slides )

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