Reversible primes

For an ‐bit positive integer written in binary as where , , , let us define the digital reversal of . Also let . With a sieve argument, we obtain an upper bound of the expected order of magnitude for the number of such that and are prime. We also prove that for sufficiently large , where denotes the number of prime factors counted with multiplicity of and is an absolute constant. Finally, we provide an asymptotic formula for the number of ‐bit integers such that and are both squarefree. Our method leads us to provide various estimates for the exponential sum