Palindromic number: Encyclopedia II - Palindromic number - Other bases

Palindromic number - Other bases

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

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.

Generally, a number that is palindromic in one base is not palindromic in another base; for instance, 16461₁₀ = 404D₁₆. (The subscripts indicate radices, so n₁₆ means n written in hexadecimal.) However, some numbers are copalindromic in several bases. The number 105₁₀, for example, is palindromic in five bases: 1221₄ = 151₈ = 77₁₄ = 55₂₀ = 33₃₄. The year 1991 is palindromic in both decimal and hexadecimal (7C7).

In base 18, some powers of seven are palindromic:

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

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 11_n−1). A number that is non-palindromic in all bases 2 ≤ b < n − 1 is called a strictly non-palindromic number.

Adapted from the Wikipedia article "Other bases", under the G.N U Free Docmentation License. Please also see http://en.wikipedia.org/wiki