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: 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.
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: with, as usual, 0 ≤ ai < b for all i and ak ≠ 0.
The other two digits are determined by the choice of the first two): 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).
Gustavus Simmons conjectured there are no palindromes of form nk for k > 4 (and n > 1).
[3] Palindromic numbers can be considered in numeral systems other than decimal.
For example, the binary palindromic numbers are those with the binary representations: or in decimal: The Fermat primes and the Mersenne primes form a subset of the binary palindromic primes.
All strictly non-palindromic numbers larger than 6 are prime.
[4][5] The first few strictly non-palindromic numbers (sequence A016038 in the OEIS) are: If the digits of a natural number don't only have to be reversed in order, but also subtracted from
to yield the original sequence again, then the number is said to be antipalindromic.
Formally, in the usual decomposition of a natural number into its digits
[6] Non-palindromic numbers can be paired with palindromic ones via a series of operations.
Such number is called "a delayed palindrome".
On January 24, 2017, the number 1,999,291,987,030,606,810 was published in OEIS as A281509 and announced "The Largest Known Most Delayed Palindrome".
The sequence of 125 261-step most delayed palindromes preceding 1,999,291,987,030,606,810 and not reported before was published separately as A281508.
The sum of the reciprocals of the palindromic numbers is a convergent series, whose value is approximately 3.37028... (sequence A118031 in the OEIS).
[7] Fuller does not give a formal definition for this term, but from the examples he gives, it can be understood to be those numbers that contain a factor of the primorial n#, where n≥13 and is the largest prime factor in the number.
Fuller also refers to powers of 1001 as Scheherazade numbers.
Fuller pointed out that some of these numbers are palindromic by groups of digits.
For instance 17# = 510,510 shows a symmetry of groups of three digits.
Fuller called such numbers Scheherazade Sublimely Rememberable Comprehensive Dividends, or SSRCD numbers.
Fuller notes that 1001 raised to a power not only produces sublimely rememberable numbers that are palindromic in three-digit groups, but also the values of the groups are the binomial coefficients.
Fuller suggests writing these spillovers on a separate line.
If this is done, using more spillover lines as necessary, the symmetry is preserved indefinitely to any power.
[8] Many other Scheherazade numbers show similar symmetries when expressed in this way.
[9] In 2018, a paper was published demonstrating that every positive integer can be written as the sum of three palindromic numbers in every number system with base 5 or greater.