palindromic number

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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 A palindrome is a word, palindromic number, number, phrase, or other sequence of symbols that reads the same backwards as forwards, such as the words ''madam'' or ''racecar'', the date and time ''11/11/11 11:11,'' and the sentence: "A man, a pla ...
, 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 The decimal numeral system (also called the base-ten positional numeral system and denary or decanary) is the standard system for denoting integer and non-integer numbers. It is the extension to non-integer numbers of the Hindu–Arabic numeral ...
) 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, … . Palindromic numbers receive most attention in the realm of
recreational mathematics Recreational mathematics is mathematics carried out for recreation (entertainment) rather than as a strictly research and application-based professional activity or as a part of a student's formal education. Although it is not necessarily limited ...
. A typical problem asks for numbers that possess a certain property ''and'' are palindromic. For instance: * The
palindromic prime In mathematics, a palindromic prime (sometimes called a palprime) is a prime number that is also a palindromic number. Palindromicity depends on the Radix, base of the number system and its notational conventions, while primality is independent of ...
s are 2, 3, 5, 7, 11, 101, 131, 151, ... . * The palindromic
square number In mathematics, a square number or perfect square is an integer that is the square (algebra), square of an integer; in other words, it is the multiplication, product of some integer with itself. For example, 9 is a square number, since it equals ...
s are 0, 1, 4, 9, 121, 484, 676, 10201, 12321, ... . It is obvious that in any base there are infinitely many palindromic numbers, since in any base the infinite
sequence In mathematics, a sequence is an enumerated collection of mathematical object, objects in which repetitions are allowed and order theory, order matters. Like a Set (mathematics), set, it contains Element (mathematics), members (also called ''eleme ...
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 The decimal numeral system (also called the base-ten positional numeral system and denary or decanary) is the standard system for denoting integer and non-integer numbers. It is the extension to non-integer numbers of the Hindu–Arabic numeral ...
system, the concept of palindromicity can be applied to the
natural numbers In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''Cardinal n ...
in any
numeral system A numeral system (or system of numeration) is a writing system for expressing numbers; that is, a mathematical notation for representing number A number is a mathematical object used to count, measure, and label. The original example ...
. Consider a number ''n'' > 0 in base ''b'' ≥ 2, where it is written in standard notation with ''k''+1 digits ''a''''i'' as: :$n=\sum_^ka_ib^i$ 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 0 (zero) is a number, and the numerical digit used to represent that number in numeral system, numerals. It fulfills a central role in mathematics as the additive identity of the integers, real numbers, and many other algebraic structures. A ...
is written 0 in any base and is also palindromic by definition.

Decimal palindromic numbers

All numbers in
base 10 The decimal numeral system (also called the base-ten positional numeral system and denary or decanary) is the standard system for denoting integer and non-integer numbers. It is the extension to non-integer numbers of the Hindu–Arabic numeral ...
(and indeed in any base) with one digit are palindromic, so there are ten decimal palindromic numbers with one digit: :. There are 9 palindromic numbers with two digits: :. 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): : 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): :, 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, ... . The number of palindromic numbers which have some other property are listed below:

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 In Euclidean geometry, a square is a regular polygon, regular quadrilateral, which means that it has four equal sides and four equal angles (90-degree (angle), degree angles, π/2 radian angles, or right angles). It can also be defined as a rec ...
: 0, 1, 4, 9, 121, 484, 676, 10201, 12321, 14641, 40804, 44944, ... * Palindromic
cubes In geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field ...
: 0, 1, 8, 343, 1331, 1030301, 1367631, 1003003001, ... * Palindromic
fourth power In arithmetic Arithmetic () is an elementary part of mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities ...
s: 0, 1, 14641, 104060401, 1004006004001, ... The first nine terms of the sequence 12, 112, 1112, 11112, ... form the palindromes 1, 121, 12321, 1234321, ... 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). G. J. Simmons conjectured there are no palindromes of form ''n''''k'' for ''k'' > 4 (and ''n'' > 1).

Other bases

Palindromic numbers can be considered in
numeral system A numeral system (or system of numeration) is a writing system for expressing numbers; that is, a mathematical notation for representing number A number is a mathematical object used to count, measure, and label. The original example ...
s other than
decimal The decimal numeral system (also called the base-ten positional numeral system and denary or decanary) is the standard system for denoting integer and non-integer numbers. It is the extension to non-integer numbers of the Hindu–Arabic numeral ...
. For example, the
binary Binary may refer to: Science and technology Mathematics * Binary number A binary number is a number expressed in the base-2 numeral system or binary numeral system, a method of mathematical expression which uses only two symbols: typical ...
palindromic numbers are those with the binary representations: :0, 1, 11, 101, 111, 1001, 1111, 10001, 10101, 11011, 11111, 100001, ... or in decimal: :0, 1, 3, 5, 7, 9, 15, 17, 21, 27, 31, 33, ... The
Fermat prime In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in m ...
s and the
Mersenne prime In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in mo ...
s form a subset of the binary palindromic primes. Any number $n$ is palindromic in all bases $b$ with $b > n$ (trivially so, because $n$ is then a single-digit number), and also in base $n-1$ (because $n$ is then $11_$). Even excluding cases where the number is smaller than the base, most numbers are palindromic in more than one base. For example, $1221_4=151_8=77_=55_=33_=11_$, $1991_=7C7_$. A number $n$ is never palindromic in base $b$ if $n/2 \le b \le n-2$. A number that is non-palindromic in all bases ''b'' in the range 2 ≤ ''b'' ≤ ''n'' − 2 can be called a ''strictly non-palindromic number''. For example, the number 6 is written as "110" in base 2, "20" in base 3, and "12" in base 4, none of which are palindromes. All strictly non-palindromic numbers larger than 6 are prime. Indeed, if $n > 6$ is composite, then either $n = ab$ for some $2 \le a < b$, in which case ''n'' is the palindrome "aa" in base $b-1$, or else it is a perfect square $n = a^2$, in which case ''n'' is the palindrome "121" in base $a-1$ (except for the special case of $n = 9 = 1001_2$).

Antipalindromic numbers

If the digits of a natural number don't only have to be reversed in order, but also subtracted from $b-1$ 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 $a_i$ in base $b$, a number is antipalindromic iff $a_i = b - 1 - a_$.

Lychrel process

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.

Sum of the reciprocals

The sum of the reciprocals of the palindromic numbers is a convergent series, whose value is approximately 3.37028... .

Scheherazade numbers are a set of numbers identified by
Buckminster Fuller Richard Buckminster Fuller (; July 12, 1895 – July 1, 1983) was an American people, American architect, systems theorist, writer, designer, inventor, philosopher, and futurist. He styled his name as R. Buckminster Fuller in his writings, ...
in his book ''Synergetics''. 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 In mathematics, and more particularly in number theory, primorial, denoted by "#", is a Function (mathematics), function from natural numbers to natural numbers similar to the factorial function, but rather than successively multiplying positive ...
''n''#, where ''n''≥13 and is the largest
prime factor A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...
in the number. Fuller called these numbers ''Scheherazade numbers'' because they must have a factor of 1001.
Scheherazade Scheherazade () is a major female character and the storyteller in the frame story, frame narrative of the Middle Eastern collection of tales known as the ''One Thousand and One Nights''. Name According to modern scholarship, the name ''Schehe ...
is the storyteller of ''
One Thousand and One Nights ''One Thousand and One Nights'' ( ar, أَلْفُ لَيْلَةٍ وَلَيْلَةٌ, italic=yes, ) is a collection of Middle Eastern folk tales compiled in Arabic during the Islamic Golden Age. It is often known in English as the ''Arabian ...
'', 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 coefficient In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers and is written \tbinom. It is the coefficient of the ter ...
s. For instance, :$\left(1001\right)^6 = 1,006,015,020,015,006,001$ 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. Many other Scheherazade numbers show similar symmetries when expressed in this way.

Sums of palindromes

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.
arXiv preprint

* Lychrel number *
Palindromic prime In mathematics, a palindromic prime (sometimes called a palprime) is a prime number that is also a palindromic number. Palindromicity depends on the Radix, base of the number system and its notational conventions, while primality is independent of ...
*
Palindrome A palindrome is a word, palindromic number, number, phrase, or other sequence of symbols that reads the same backwards as forwards, such as the words ''madam'' or ''racecar'', the date and time ''11/11/11 11:11,'' and the sentence: "A man, a pla ...

References

*Malcolm E. Lines: ''A Number for Your Thoughts: Facts and Speculations about Number from Euclid to the latest Computers'': CRC Press 1986, , S. 61