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" | $313$
 * align="right" style = "padding: 2px 10px" | $100 \, 111 \, 001$
 * }
 * }