Reversible primes,Journal of the London Mathematical Society - X-MOL

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Reversible primes
Journal of the London Mathematical Society ( IF 1.2 ) Pub Date : 2024-03-07 , DOI: 10.1112/jlms.12883
Cécile Dartyge ₁ , Bruno Martin ₂ , Joël Rivat ₃ , Igor E. Shparlinski ₄ , Cathy Swaenepoel ₅

Affiliation

For an

n $n$

-bit positive integer

a $a$

written in binary as

a = \sum_{j = 0}^{n - 1} ε_{j} (a) 2^{j}, $ {a} = \sum _{j=0}^{n-1} \varepsilon _{j}(a) \,2^j,$

where

ε_{j} (a) \in {0, 1} $\varepsilon _j(a) \in \lbrace 0,1\rbrace$

,

j \in {0, …, n - 1} $j\in \lbrace 0, \ldots , n-1\rbrace$

,

ε_{n - 1} (a) = 1 $\varepsilon _{n-1}(a)=1$

, let us define

\overset{\leftarrow}{a} = \sum_{j = 0}^{n - 1} ε_{j} (a) 2^{n - 1 - j}, $ \overleftarrow{a} = \sum _{j=0}^{n-1} \varepsilon _j(a)\,2^{n-1-j},$

the digital reversal of

a $a$

. Also let

B_{n} = {2^{n - 1} ⩽ a < 2^{n} : a odd} ${\mathcal {B}}_n= \lbrace 2^{n-1}\leqslant a<2^n:\nobreakspace a \text{ odd}\rbrace$

. With a sieve argument, we obtain an upper bound of the expected order of magnitude for the number of

p \in B_{n} $p \in {\mathcal {B}}_n$

such that

p $p$

and

\overset{\leftarrow}{p} $\overleftarrow{p}$

are prime. We also prove that for sufficiently large

n $n$

,

更新日期：2024-03-10

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