Palindromes in Base 10 and Base 3

Theorem

The following $n \in \Z$ are palindromic in both decimal and ternary:

$0, 1, 2, 4, 8, 121, 151, 212, 242, 484, 656, 757, \ldots$

This sequence is A007633 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).

Proof

$n_{10}$	$n_3$
$0$	$0$
$1$	$1$
$2$	$2$
$4$	$11$
$8$	$22$
$121$	$11 \, 111$
$151$	$12 \, 121$
$212$	$21 \, 212$
$242$	$22 \, 222$
$484$	$122 \, 221$
$656$	$220 \, 022$
$757$	$1 \, 001 \, 001$

$\blacksquare$

Sources

• 1985: J. Meeus: Multibasic Palindromes (J. Recr. Math. Vol. 18, no. 3: pp. 168 – 173)

• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $121$