Palindromes in Base 10 and Base 2
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Theorem
The following $n \in \Z$ are palindromic in both decimal and binary:
- $0, 1, 3, 5, 7, 9, 33, 99, 313, 585, 717, 7447, 9009, 15 \, 351, 32 \, 223, 39 \, 993, \ldots$
This sequence is A007632 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).
Proof
$n_{10}$ $n_2$ $0$ $0$ $1$ $1$ $3$ $11$ $5$ $101$ $7$ $111$ $9$ $1001$ $33$ $100 \, 001$ $99$ $1 \, 100 \, 011$ $313$ $100 \, 111 \, 001$ $585$ $1 \, 001 \, 001 \, 001$ $717$ $1 \, 011 \, 001 \, 101$ $7447$ $1 \, 110 \, 100 \, 010 \, 111$ $15 \, 351$ $11 \, 101 \, 111 \, 110 \, 111$ $32 \, 223$ $111 \, 110 \, 111 \, 011 \, 111$ $39 \, 993$ $1 \, 001 \, 110 \, 000 \, 111 \, 001$
$\blacksquare$
Sources
- 1985: M.R. Calandra: Integers which are Palindromic in both Decimal and Binary Notation (J. Recr. Math. Vol. 18, no. 1: p. 47)
- 1985: S. Pilpel: Some More Double Palindromic Integers (J. Recr. Math. Vol. 18: pp. 174 – 176)
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $33$