Palindromic Number - Other Bases

Other Bases

Palindromic numbers can be considered in other numeral systems than decimal. For example, the binary palindromic numbers are:

0, 1, 11, 101, 111, 1001, 1111, 10001, 10101, 11011, 11111, 100001, …

or in decimal: 0, 1, 3, 5, 7, 9, 15, 17, 21, 27, 31, 33, … (sequence A006995 in OEIS). The Mersenne primes form a subset of the binary palindromic primes.

All numbers are palindromic in an infinite number of bases. But, it's more interesting to consider bases smaller than the number itself - in which case most numbers are palindromic in more than one base.

In base 18, some powers of seven are palindromic:

73 = 111 74 = 777 76 = 12321 79 = 1367631

And in base 24 the first eight powers of five are palindromic as well:

51 = 5 52 = 11 53 = 55 54 = 121 55 = 5A5 56 = 1331 57 = 5FF5 58 = 14641 5A = 15AA51 5C = 16FLF61

Any number n is palindromic in all bases b with b ≥ n + 1 (trivially so, because n is then a single-digit number), and also in base n−1 (because n is then 11n−1). A number that is non-palindromic in all bases 2 ≤ b < n − 1 is called a strictly non-palindromic number.

Read more about this topic:  Palindromic Number

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