Palindromic number
Encyclopedia
A palindromic number or numeral palindrome is a 'symmetrical' number like 16461, that remains the same when its digits are reversed. The term palindromic is derived from palindrome
, which refers to a word like rotor that remains unchanged under reversal of its letters. The first 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. For instance:
Buckminster Fuller
referred to palindromic numbers as Scheherazade numbers in his book Synergetics
, because Scheherazade
was the name of the story-telling wife in the 1001 Nights
.
It is fairly straightforward to appreciate 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, etc. (in which the nth number is a 1, followed by n zeros, followed by a 1) consists of palindromic numbers only.
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 digit
s ai as:
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.
with one digit
are palindromic. The number of palindromic numbers with two digits is 9:
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):
and also 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, … . For some types of palindromic numbers these values are listed below in a table. Here 0 is included.
No palindromes of form n5 (or higher exponent) have been found.
The only known non-palindromic number, whose cube is a palindrome, is 2201.
G. J. Simmons conjectured there are no palindromes of form nk for k > 4.
s than decimal
. For example, the binary
palindromic numbers are:
or in decimal: 0, 1, 3, 5, 7, 9, 15, 17, 21, 27, 31, 33, … . The Mersenne prime
s form a subset of the binary palindromic primes.
All numbers are palindromic in an infinite number of bases. But, it's more interesting to consider bases smaller than the number itself - in which case most numbers are palindromic in more than one base.
In base 18, some powers of seven are palindromic:
73 = 111
74 = 777
76 = 12321
79 = 1367631
And in base 24
the first eight powers of five are palindromic as well:
51 = 5
52 = 11
53 = 55
54 = 121
55 = 5A5
56 = 1331
57 = 5FF5
58 = 14641
5A = 15AA51
5C = 16FLF61
Any number n is palindromic in all bases b with b ≥ n + 1 (trivially so, because n is then a single-digit number), and also in base n−1 (because n is then 11n−1). A number that is non-palindromic in all bases 2 ≤; b < n − 1 is called a strictly non-palindromic number
.
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
.
Palindrome
A palindrome is a word, phrase, number, or other sequence of units that can be read the same way in either direction, with general allowances for adjustments to punctuation and word dividers....
, which refers to a word like rotor that remains unchanged under reversal of its letters. The first palindromic numbers (in decimal
Decimal
The decimal numeral system has ten as its base. It is the numerical base most widely used by modern civilizations....
) 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, … .
Palindromic numbers receive most attention in the realm of recreational mathematics
Recreational mathematics
Recreational mathematics is an umbrella term, referring to mathematical puzzles and mathematical games.Not all problems in this field require a knowledge of advanced mathematics, and thus, recreational mathematics often attracts the curiosity of non-mathematicians, and inspires their further study...
. A typical problem asks for numbers that possess a certain property and are palindromic. For instance:
- The palindromic primePalindromic primeA palindromic prime is a prime number that is also a palindromic number. Palindromicity depends on the base of the numbering system and its writing conventions, while primality is independent of such concerns...
s are 2, 3, 5, 7, 11, 101, 131, 151, … . - The palindromic square numberSquare numberIn mathematics, a square number, sometimes also called a perfect square, is an integer that is the square of an integer; in other words, it is the product of some integer with itself...
s are 0, 1, 4, 9, 121, 484, 676, 10201, 12321, … .
Buckminster Fuller
Buckminster Fuller
Richard Buckminster “Bucky” Fuller was an American systems theorist, author, designer, inventor, futurist and second president of Mensa International, the high IQ society....
referred to palindromic numbers as Scheherazade numbers in his book Synergetics
Synergetics (Fuller)
Synergetics is the empirical study of systems in transformation, with an emphasis on total system behavior unpredicted by the behavior of any isolated components, including humanity’s role as both participant and observer....
, because Scheherazade
Scheherazade
Scheherazade , sometimes Scheherazadea, Persian transliteration Shahrazad or Shahrzād is a legendary Persian queen and the storyteller of One Thousand and One Nights.-Narration :...
was the name of the story-telling wife in the 1001 Nights
The Book of One Thousand and One Nights
One Thousand and One Nights is a collection of Middle Eastern and South Asian stories and folk tales compiled in Arabic during the Islamic Golden Age...
.
It is fairly straightforward to appreciate that in any base there are infinitely many palindromic numbers, since in any base the infinite sequence
Sequence
In mathematics, a sequence is an ordered list of objects . Like a set, it contains members , and the number of terms is called the length of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence...
of numbers written (in that base) as 101, 1001, 10001, etc. (in which the nth number is a 1, followed by n zeros, followed by a 1) consists of palindromic numbers only.
Formal definition
Although palindromic numbers are most often considered in the decimalDecimal
The decimal numeral system has ten as its base. It is the numerical base most widely used by modern civilizations....
system, the concept of palindromicity can be applied to the natural numbers in any numeral system
Numeral system
A numeral system is a writing system for expressing numbers, that is a mathematical notation for representing numbers of a given set, using graphemes or symbols in a consistent manner....
. Consider a number n > 0 in base
Radix
In mathematical numeral systems, the base or radix for the simplest case is the number of unique digits, including zero, that a positional numeral system uses to represent numbers. For example, for the decimal system the radix is ten, because it uses the ten digits from 0 through 9.In any numeral...
b ≥ 2, where it is written in standard notation with k+1 digit
Numerical digit
A digit is a symbol used in combinations to represent numbers in positional numeral systems. The name "digit" comes from the fact that the 10 digits of the hands correspond to the 10 symbols of the common base 10 number system, i.e...
s ai as:
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
0 (number)
0 is both a numberand the numerical digit used to represent that number in numerals.It fulfills a central role in mathematics as the additive identity of the integers, real numbers, and many other algebraic structures. As a digit, 0 is used as a placeholder in place value systems...
is written 0 in any base and is also palindromic by definition.
Decimal palindromic numbers
All numbers in base 10Decimal
The decimal numeral system has ten as its base. It is the numerical base most widely used by modern civilizations....
with one digit
Numerical digit
A digit is a symbol used in combinations to represent numbers in positional numeral systems. The name "digit" comes from the fact that the 10 digits of the hands correspond to the 10 symbols of the common base 10 number system, i.e...
are palindromic. The number of palindromic numbers with two digits is 9:
- {11, 22, 33, 44, 55, 66, 77, 88, 99}.
There are 90 palindromic numbers with three digits (Using the Rule of product
Rule of product
In combinatorics, the rule of product or multiplication principle is a basic counting principle...
: 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}
and also 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, … . For some types of palindromic numbers these values are listed below in a table. Here 0 is included.
| 101 | 102 | 103 | 104 | 105 | 106 | 107 | 108 | 109 | 1010 | |
| n natural Natural number In mathematics, the natural numbers are the ordinary whole numbers used for counting and ordering . These purposes are related to the linguistic notions of cardinal and ordinal numbers, respectively... | 10 | 19 | 109 | 199 | 1099 | 1999 | 10999 | 19999 | 109999 | 199999 |
| n even Even and odd numbers In mathematics, the parity of an object states whether it is even or odd.This concept begins with integers. An even number is an integer that is "evenly divisible" by 2, i.e., divisible by 2 without remainder; an odd number is an integer that is not evenly divisible by 2... | 5 | 9 | 49 | 89 | 489 | 889 | 4889 | 8889 | 48889 | 88889 |
| n odd | 5 | 10 | 60 | 110 | 610 | 1110 | 6110 | 11110 | 61110 | 111110 |
| n square Square number In mathematics, a square number, sometimes also called a perfect square, is an integer that is the square of an integer; in other words, it is the product of some integer with itself... | 4 | 7 | 14 | 15 | 20 | 31 | ||||
| n cube | 3 | 4 | 5 | 7 | 8 | |||||
| n prime Prime number A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2... | 4 | 5 | 20 | 113 | 781 | 5953 | ||||
| n squarefree Square-free integer In mathematics, a square-free, or quadratfrei, integer is one divisible by no perfect square, except 1. For example, 10 is square-free but 18 is not, as it is divisible by 9 = 32... | 6 | 12 | 67 | 120 | 675 | 1200 | 6821 | 12160 | + | + |
| n non-squarefree (μ(n) Möbius function The classical Möbius function μ is an important multiplicative function in number theory and combinatorics. The German mathematician August Ferdinand Möbius introduced it in 1832... =0) | 4 | 7 | 42 | 79 | 424 | 799 | 4178 | 7839 | + | + |
| n square with prime root | 2 | 3 | 5 | |||||||
| n with an even number of distinct prime factor Prime factor In number theory, the prime factors of a positive integer are the prime numbers that divide that integer exactly, without leaving a remainder. The process of finding these numbers is called integer factorization, or prime factorization. A prime factor can be visualized by understanding Euclid's... s (μ(n)=1) | 2 | 6 | 35 | 56 | 324 | 583 | 3383 | 6093 | + | + |
| n with an odd number of distinct prime factors (μ(n)=-1) | 4 | 6 | 32 | 64 | 351 | 617 | 3438 | 6067 | + | + |
| n even with an odd number of prime factors | 1 | 2 | 9 | 21 | 100 | 180 | 1010 | 6067 | + | + |
| n even with an odd number of distinct prime factors | 3 | 4 | 21 | 49 | 268 | 482 | 2486 | 4452 | + | + |
| n odd with an odd number of prime factors | 3 | 4 | 23 | 43 | 251 | 437 | 2428 | 4315 | + | + |
| n odd with an odd number of distinct prime factors | 4 | 5 | 28 | 56 | 317 | 566 | 3070 | 5607 | + | + |
| n even squarefree with an even number of (distinct) prime factors | 1 | 2 | 11 | 15 | 98 | 171 | 991 | 1782 | + | + |
| n odd squarefree with an even number of (distinct) prime factors | 1 | 4 | 24 | 41 | 226 | 412 | 2392 | 4221 | + | + |
| n odd with exactly 2 prime factors | 1 | 4 | 25 | 39 | 205 | 303 | 1768 | 2403 | + | + |
| n even with exactly 2 prime factors | 2 | 3 | 11 | 64 | 413 | + | + | |||
| n even with exactly 3 prime factors | 1 | 3 | 14 | 24 | 122 | 179 | 1056 | 1400 | + | + |
| n even with exactly 3 distinct prime factors | 0 | 1 | 18 | 44 | 250 | 390 | 2001 | 2814 | + | + |
| n odd with exactly 3 prime factors | 0 | 1 | 12 | 34 | 173 | 348 | 1762 | 3292 | + | + |
| n Carmichael number | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 |
| n for which σ(n) Divisor function In mathematics, and specifically in number theory, a divisor function is an arithmetical function related to the divisors of an integer. When referred to as the divisor function, it counts the number of divisors of an integer. It appears in a number of remarkable identities, including relationships... is palindromic | 6 | 10 | 47 | 114 | 688 | 1417 | 5683 | + | + | + |
Perfect powers
There are many palindromic perfect powers nk, where n is a natural number and k is 2, 3 or 4.- Palindromic squaresSquare numberIn mathematics, a square number, sometimes also called a perfect square, is an integer that is the square of an integer; in other words, it is the product of some integer with itself...
: 0, 1, 4, 9, 121, 484, 676, 10201, 12321, 14641, 40804, 44944, ... - Palindromic cubes: 0, 1, 8, 343, 1331, 1030301, 1367631, 1003003001, ...
- Palindromic fourth powersFourth powerIn arithmetic and algebra, the fourth power of a number n is the result of multiplying n by itself four times. So:Fourth powers are also formed by multiplying a number by its cube...
: 0, 1, 14641, 104060401, 1004006004001, ...
No palindromes of form n5 (or higher exponent) have been found.
The only known non-palindromic number, whose cube is a palindrome, is 2201.
G. J. Simmons conjectured there are no palindromes of form nk for k > 4.
Other bases
Palindromic numbers can be considered in other numeral systemNumeral system
A numeral system is a writing system for expressing numbers, that is a mathematical notation for representing numbers of a given set, using graphemes or symbols in a consistent manner....
s than decimal
Decimal
The decimal numeral system has ten as its base. It is the numerical base most widely used by modern civilizations....
. For example, the binary
Binary numeral system
The binary numeral system, or base-2 number system, represents numeric values using two symbols, 0 and 1. More specifically, the usual base-2 system is a positional notation with a radix of 2...
palindromic numbers are:
- 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 Mersenne prime
Mersenne prime
In mathematics, a Mersenne number, named after Marin Mersenne , is a positive integer that is one less than a power of two: M_p=2^p-1.\,...
s form a subset of the binary palindromic primes.
All numbers are palindromic in an infinite number of bases. But, it's more interesting to consider bases smaller than the number itself - in which case most numbers are palindromic in more than one base.
In base 18, some powers of seven are palindromic:
73 = 111
74 = 777
76 = 12321
79 = 1367631
And in base 24
Base 24
The base- system is a numeral system with 24 as its base.There are 24 hours in a nychthemeron , so solar time includes a base-24 component.See also base 12. Decimal Equivalent...
the first eight powers of five are palindromic as well:
51 = 5
52 = 11
53 = 55
54 = 121
55 = 5A5
56 = 1331
57 = 5FF5
58 = 14641
5A = 15AA51
5C = 16FLF61
Any number n is palindromic in all bases b with b ≥ n + 1 (trivially so, because n is then a single-digit number), and also in base n−1 (because n is then 11n−1). A number that is non-palindromic in all bases 2 ≤; b < n − 1 is called a strictly non-palindromic number
Strictly non-palindromic number
A strictly non-palindromic number is an integer n that is not palindromic in any numeral system with a base b in the range 2 ≤ b ≤ n − 2...
.
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.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
Lychrel number
A Lychrel number is a natural number which cannot form a palindrome through the iterative process of repeatedly reversing its base 10 digits and adding the resulting numbers. This process is called the 196-algorithm. The name "Lychrel" was coined by Wade VanLandingham—a rough anagram of his...
.