Palindromes in Base 10 and Base 2

Theorem
The following $n \in \Z$ are palindromic in both decimal and binary:
 * $0, 1, 3, 5, 7, 9, 33, 99, 313, 585, 717, 7447, 9009, 15 \, 351, 32 \, 223, 39 \, 993, \ldots$

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" | $0$
 * align="right" style = "padding: 2px 10px" | $0$
 * align="right" style = "padding: 2px 10px" | $1$
 * align="right" style = "padding: 2px 10px" | $1$
 * 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" | $9$
 * align="right" style = "padding: 2px 10px" | $1001$
 * align="right" style = "padding: 2px 10px" | $33$
 * align="right" style = "padding: 2px 10px" | $100 \, 001$
 * align="right" style = "padding: 2px 10px" | $99$
 * align="right" style = "padding: 2px 10px" | $1 \, 100 \, 011$
 * align="right" style = "padding: 2px 10px" | $313$
 * align="right" style = "padding: 2px 10px" | $100 \, 111 \, 001$
 * align="right" style = "padding: 2px 10px" | $585$
 * align="right" style = "padding: 2px 10px" | $1 \, 001 \, 001 \, 001$
 * align="right" style = "padding: 2px 10px" | $717$
 * align="right" style = "padding: 2px 10px" | $1 \, 011 \, 001 \, 101$
 * align="right" style = "padding: 2px 10px" | $7447$
 * align="right" style = "padding: 2px 10px" | $1 \, 110 \, 100 \, 010 \, 111$
 * align="right" style = "padding: 2px 10px" | $15 \, 351$
 * align="right" style = "padding: 2px 10px" | $11 \, 101 \, 111 \, 110 \, 111$
 * align="right" style = "padding: 2px 10px" | $32 \, 223$
 * align="right" style = "padding: 2px 10px" | $111 \, 110 \, 111 \, 011 \, 111$
 * align="right" style = "padding: 2px 10px" | $39 \, 993$
 * align="right" style = "padding: 2px 10px" | $1 \, 001 \, 110 \, 000 \, 111 \, 001$
 * }
 * align="right" style = "padding: 2px 10px" | $717$
 * align="right" style = "padding: 2px 10px" | $1 \, 011 \, 001 \, 101$
 * align="right" style = "padding: 2px 10px" | $7447$
 * align="right" style = "padding: 2px 10px" | $1 \, 110 \, 100 \, 010 \, 111$
 * align="right" style = "padding: 2px 10px" | $15 \, 351$
 * align="right" style = "padding: 2px 10px" | $11 \, 101 \, 111 \, 110 \, 111$
 * align="right" style = "padding: 2px 10px" | $32 \, 223$
 * align="right" style = "padding: 2px 10px" | $111 \, 110 \, 111 \, 011 \, 111$
 * align="right" style = "padding: 2px 10px" | $39 \, 993$
 * align="right" style = "padding: 2px 10px" | $1 \, 001 \, 110 \, 000 \, 111 \, 001$
 * }
 * align="right" style = "padding: 2px 10px" | $39 \, 993$
 * align="right" style = "padding: 2px 10px" | $1 \, 001 \, 110 \, 000 \, 111 \, 001$
 * }
 * }