Palindromic Primes in Base 10 and Base 2

Theorem
The following $n \in \Z$ are prime numbers which are palindromic in both decimal and binary:
 * $3, 5, 7, 313, 7 \, 284 \, 717 \, 174 \, 827, 390 \, 714 \, 505 \, 091 \, 666 \, 190 \, 505 \, 417 \, 093, \ldots$

It is not known whether there are any more.

Proof

 * {| border="1"

! align="center" style = "padding: 2px 10px" | $n_{10}$ ! align="center" style = "padding: 2px 10px" | $n_2$
 * align="right" style = "padding: 2px 10px" | $3$
 * align="right" style = "padding: 2px 10px" | $11$
 * align="right" style = "padding: 2px 10px" | $5$
 * align="right" style = "padding: 2px 10px" | $101$
 * align="right" style = "padding: 2px 10px" | $7$
 * align="right" style = "padding: 2px 10px" | $111$
 * align="right" style = "padding: 2px 10px" | $313$
 * align="right" style = "padding: 2px 10px" | $100 \, 111 \, 001$
 * align="right" style = "padding: 2px 10px" | $7 \, 284 \, 717 \, 174 \, 827$
 * align="right" style = "padding: 2px 10px" | $1 \, 101 \, 010 \, 000 \, 000 \, 011 \, 010 \, 111 \, 110 \, 101 \, 100 \, 000 \, 000 \, 101 \, 011$
 * align="right" style = "padding: 2px 10px" | $390 \, 714 \, 505 \, 091 \, 666 \, 190 \, 505 \, 417 \, 093$
 * align="right" style = "padding: 2px 10px" | $10 \, 100 \, 001 \, 100 \, 110 \, 001 \, 000 \, 000 \, 111 \, 100 \, 001 \, 100 \, 011 \, 000 \, 111 \, 011 \, 100 \, 011 \, 000 \, 110 \, 000 \, 111 \, 100 \, 000 \, 010 \, 001 \, 100 \, 110 \, 000 \, 101$
 * }
 * align="right" style = "padding: 2px 10px" | $7 \, 284 \, 717 \, 174 \, 827$
 * align="right" style = "padding: 2px 10px" | $1 \, 101 \, 010 \, 000 \, 000 \, 011 \, 010 \, 111 \, 110 \, 101 \, 100 \, 000 \, 000 \, 101 \, 011$
 * align="right" style = "padding: 2px 10px" | $390 \, 714 \, 505 \, 091 \, 666 \, 190 \, 505 \, 417 \, 093$
 * align="right" style = "padding: 2px 10px" | $10 \, 100 \, 001 \, 100 \, 110 \, 001 \, 000 \, 000 \, 111 \, 100 \, 001 \, 100 \, 011 \, 000 \, 111 \, 011 \, 100 \, 011 \, 000 \, 110 \, 000 \, 111 \, 100 \, 000 \, 010 \, 001 \, 100 \, 110 \, 000 \, 101$
 * }
 * }

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