Palindromes in Base 10 and Base 3

Theorem
The following $n \in \Z$ are palindromic in both decimal and ternary:
 * $0, 1, 2, 4, 8, 121, 151, 212, 242, 484, 656, 757, \ldots$

Proof

 * {| border="1"

! align="center" style = "padding: 2px 10px" | $n_{10}$ ! align="center" style = "padding: 2px 10px" | $n_3$
 * 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" | $2$
 * align="right" style = "padding: 2px 10px" | $2$
 * align="right" style = "padding: 2px 10px" | $4$
 * align="right" style = "padding: 2px 10px" | $11$
 * align="right" style = "padding: 2px 10px" | $8$
 * align="right" style = "padding: 2px 10px" | $22$
 * align="right" style = "padding: 2px 10px" | $121$
 * align="right" style = "padding: 2px 10px" | $11 \, 111$
 * align="right" style = "padding: 2px 10px" | $151$
 * align="right" style = "padding: 2px 10px" | $12 \, 121$
 * align="right" style = "padding: 2px 10px" | $212$
 * align="right" style = "padding: 2px 10px" | $21 \, 212$
 * align="right" style = "padding: 2px 10px" | $242$
 * align="right" style = "padding: 2px 10px" | $22 \, 222$
 * align="right" style = "padding: 2px 10px" | $484$
 * align="right" style = "padding: 2px 10px" | $122 \, 221$
 * align="right" style = "padding: 2px 10px" | $656$
 * align="right" style = "padding: 2px 10px" | $220 \, 022$
 * align="right" style = "padding: 2px 10px" | $757$
 * align="right" style = "padding: 2px 10px" | $1 \, 001 \, 001$
 * }
 * align="right" style = "padding: 2px 10px" | $242$
 * align="right" style = "padding: 2px 10px" | $22 \, 222$
 * align="right" style = "padding: 2px 10px" | $484$
 * align="right" style = "padding: 2px 10px" | $122 \, 221$
 * align="right" style = "padding: 2px 10px" | $656$
 * align="right" style = "padding: 2px 10px" | $220 \, 022$
 * align="right" style = "padding: 2px 10px" | $757$
 * align="right" style = "padding: 2px 10px" | $1 \, 001 \, 001$
 * }
 * align="right" style = "padding: 2px 10px" | $757$
 * align="right" style = "padding: 2px 10px" | $1 \, 001 \, 001$
 * }