A palindromic number (also known as a numeral palindrome or a numeric palindrome) is a number (such as 16361) 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). 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.
Formal definition Although palindromic numbers are most often considered in the decimal system, the concept of palindromicity can be applied to the natural numbers in any numeral system. Consider a number n > 0 in base b ≥ 2, where it is written in standard notation with k+1 digits ai as:
n = ∑ i = 0 k a i b i {\displaystyle n=\sum _{i=0}^{k}a_{i}b^{i}}
with, as usual, 0 ≤ ai < b for all i and ak ≠ 0. Then n is palindromic if and only if ai = ak−i for all i. Zero is written 0 in any base and is also palindromic by definition.
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. Similarly, if equal-interval multiples of 3 are half of 6-digit palindromes (e.g. 345,543 and 852,258), all such palindromes (6-digit, 12-digit, 18-digit and so on) are multiples of 1,221 (11 × 111). 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). The number of palindromic numbers which have some other property are listed below:
Perfect powers There are many palindromic perfect powers nk, 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). Gustavus Simmons conjectured there are no palindromes of form nk for k > 4 (and n > 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:
… excerpt ends here. Continue reading the full article.
