Palindromic number

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 a_i as:

${\displaystyle n=\sum _{i=0}^{k}a_{i}b^{i}}$

with, as usual, 0 ≤ a_i < b for all i and a_k ≠ 0. Then n is palindromic if and only if a_i = a_k−i for all i. Zero is written 0 in any base and is also palindromic by definition.

All numbers in base 10 (and indeed in any base) with one digit are palindromic, so there are ten decimal 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}.

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 below 10⁴.

Below 10⁵ there are 1099 palindromic numbers 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:

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

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.

Any number ${\displaystyle n}$ is palindromic in all bases ${\displaystyle b}$ with ${\displaystyle b>n}$ (trivially so, because ${\displaystyle n}$ is then a single-digit number), and also in base ${\displaystyle n-1}$ (because ${\displaystyle n}$ is then ${\displaystyle 11_{n-1}}$). Even excluding cases where the number is smaller than the base, most numbers are palindromic in more than one base. For example, ${\displaystyle 1221_{4}=151_{8}=77_{14}=55_{20}=33_{34}=11_{104}}$, ${\displaystyle 1991_{10}=7C7_{16}}$. A number that is non-palindromic in all bases ${\displaystyle b}$ where ${\displaystyle 1<b<n-1}$ is called a strictly non-palindromic number.

In base 7, because 101₇ is twice a perfect square (5²=34₇), several of its multiples are palindromic squares:

In base 18, some powers of seven are palindromic:

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

A palindromic number in base b that is made up of palindromic sequences of length l arranged in a palindromic order (such as 101 111 010 111 101₂) is palindromic in base b^l (for example the above binary number is palindromic in base 2³=8 (it is equal to 57275₈))

The square of 133₁₀ in base 30 is 4D₃₀² = KKK₃₀ = 3R₃₆² = DPD₃₆. In base 24 there are more palindromic squares due to 5² = 11. And squares of all numbers in the form 1666...6667 (with any number of 6'es between the 1 and 7) are palindromic. 167² = 1E5E1, 1667² = 1E3K3E1, 16667² = 1E3H8H3E1.

Non-palindromic numbers can be paired with palindromic ones via a series of operations. First, the non-palindromic number is reversed and the result is added to the original number. If the result is not a palindromic number, this is repeated until it gives a palindromic number. Such number is called "a delayed palindrome".

It is not known whether all non-palindromic numbers can be paired with palindromic numbers in this way. While no number has been proven to be unpaired, many do not appear to be. For example, 196 does not yield a palindrome even after 700,000,000 iterations. Any number that never becomes palindromic in this way is known as a Lychrel number.

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).

Scheherazade numbers are a set of numbers identified by Buckminster Fuller in his book Synergetics.[3] 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 called these numbers Scheherazade numbers because they must have a factor of 1001. Scheherazade is the storyteller of One Thousand and One Nights, telling a new story each night to delay her execution. Since n must be at least 13, the primorial must be at least 1·2·3·5·7·11·13, and 7×11×13 = 1001. Fuller also refers to powers of 1001 as Scheherazade numbers. The smallest primorial containing Scheherazade number is 13# = 30,030.

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. For instance,

${\displaystyle (1001)^{6}=1,006,015,020,015,006,001}$

This sequence fails at (1001)¹³ because there is a carry digit taken into the group to the left in some groups. 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.[4] Many other Scheherazade numbers show similar symmetries when expressed in this way.[5]

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.[6]

• Lychrel number
• Palindromic prime
• Palindrome
• Strictly non-palindromic number

• Malcolm E. Lines: A Number for Your Thoughts: Facts and Speculations about Number from Euclid to the latest Computers: CRC Press 1986, ISBN 0-85274-495-1, S. 61 (Limited Online-Version (Google Books))

• Weisstein, Eric W. "Palindromic Number". MathWorld.
• Jason Doucette - 196 Palindrome Quest / Most Delayed Palindromic Number
• 196 and Other Lychrel Numbers
• On General Palindromic Numbers at MathPages
• Palindromic Numbers to 100,000 from Ask Dr. Math
• P. De Geest, Palindromic cubes
• Yutaka Nishiyama, Numerical Palindromes and the 196 Problem, IJPAM, Vol.80, No.3, 375-384, 2012.