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): $121$
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