Palindromic number

A palindromic number (also known as a numeral palindrome or a numeric palindrome) is a number (such as 16461) that remains the same when its digits are reversed.

In other words, it has reflectional symmetry across a vertical axis.

The term palindromic is derived from palindrome, which refers to a word (such as rotor or racecar) whose spelling is unchanged when its letters are reversed.

The first 30 palindromic numbers (in decimal) are:

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 22, 33, 44, 55, 66, 77, 88, 99, 101, 111, 121, 131, 141, 151, 161, 171, 181, 191, 202, ... (sequence A002113 in the OEIS).

Palindromic numbers receive most attention in the realm of recreational mathematics.

A typical problem asks for numbers that possess a certain property and are palindromic.

For instance:

The palindromic primes are 2, 3, 5, 7, 11, 101, 131, 151, ... (sequence A002385 in the OEIS).

The palindromic square numbers are 0, 1, 4, 9, 121, 484, 676, 10201, 12321, ... (sequence A002779 in the OEIS).

It is obvious that in any base there are infinitely many palindromic numbers, since in any base the infinite sequence of numbers written (in that base) as 101, 1001, 10001, 100001, etc. consists solely of palindromic numbers.

Decimal palindromic numbers

All numbers with one digit are palindromic, so in base 10 there are ten palindromic numbers with one digit:

{0, 1, 2, 3, 4, 5, 6, 7, 8, 9}.

There are 9 palindromic numbers with two digits:

{11, 22, 33, 44, 55, 66, 77, 88, 99}.

All palindromic numbers with an even number of digits are divisible by 11.

There are 90 palindromic numbers with three digits (Using the rule of product: 9 choices for the first digit - which determines the third digit as well - multiplied by 10 choices for the second digit):

{101, 111, 121, 131, 141, 151, 161, 171, 181, 191, ..., 909, 919, 929, 939, 949, 959, 969, 979, 989, 999}

There are likewise 90 palindromic numbers with four digits (again, 9 choices for the first digit multiplied by ten choices for the second digit. The other two digits are determined by the choice of the first two):

{1001, 1111, 1221, 1331, 1441, 1551, 1661, 1771, 1881, 1991, ..., 9009, 9119, 9229, 9339, 9449, 9559, 9669, 9779, 9889, 9999},

so there are 199 palindromic numbers smaller than 10⁴.

There are 1099 palindromic numbers smaller than 105 and for other exponents of 10ⁿ we have: 1999, 10999, 19999, 109999, 199999, 1099999, ... (sequence A070199 in the OEIS).

Perfect powers

There are many palindromic perfect powers n^k, where n is a natural number and k is 2, 3 or 4.

Palindromic squares: 0, 1, 4, 9, 121, 484, 676, 10201, 12321, 14641, 40804, 44944, ... (sequence A002779 in the OEIS)

Palindromic cubes: 0, 1, 8, 343, 1331, 1030301, 1367631, 1003003001, ... (sequence A002781 in the OEIS)

Palindromic fourth powers: 0, 1, 14641, 104060401, 1004006004001, ... (sequence A186080 in the OEIS)

The first nine terms of the sequence 1², 11², 111², 1111², ... form the palindromes 1, 121, 12321, 1234321, ... (sequence A002477 in the OEIS)

The only known non-palindromic number whose cube is a palindrome is 2201, and it is a conjecture the fourth root of all the palindrome fourth powers are a palindrome with 100000...000001 (10ⁿ + 1).

Other bases

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

0, 1, 11, 101, 111, 1001, 1111, 10001, 10101, 11011, 11111, 100001, ... (sequence A057148 in the OEIS)

or in decimal:

0, 1, 3, 5, 7, 9, 15, 17, 21, 27, 31, 33, ... (sequence A006995 in the OEIS)

The Fermat primes and the Mersenne primes form a subset of the binary palindromic primes.

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