Palindromic Primes in Base 10 and Base 2

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Theorem

The following $n \in \Z$ are prime numbers which are palindromic in both decimal and binary:

$3, 5, 7, 313, 7 \, 284 \, 717 \, 174 \, 827, 390 \, 714 \, 505 \, 091 \, 666 \, 190 \, 505 \, 417 \, 093, \ldots$

This sequence is A046472 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).


It is not known whether there are any more.


Proof

$n_{10}$ $n_2$
$3$ $11$
$5$ $101$
$7$ $111$
$313$ $100 \, 111 \, 001$
$7 \, 284 \, 717 \, 174 \, 827$ $1 \, 101 \, 010 \, 000 \, 000 \, 011 \, 010 \, 111 \, 110 \, 101 \, 100 \, 000 \, 000 \, 101 \, 011$
$390 \, 714 \, 505 \, 091 \, 666 \, 190 \, 505 \, 417 \, 093$ $10 \, 100 \, 001 \, 100 \, 110 \, 001 \, 000 \, 000 \, 111 \, 100 \, 001 \, 100 \, 011 \, 000 \, 111 \, 011 \, 100 \, 011 \, 000 \, 110 \, 000 \, 111 \, 100 \, 000 \, 010 \, 001 \, 100 \, 110 \, 000 \, 101$


The last two numbers have $43$ and $89$ digits (in binary) respectively .

$\blacksquare$


Sources