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Encyclopedia > Palindromic number
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A palindromic number 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. Jump to: navigation, search A palindrome is a word, phrase, number or any other sequence of units (like a strand of DNA) which has the property of reading the same in either direction (the adjustment of spaces between letters is generally permitted). ...


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, Recreational mathematics includes many mathematical games, and can be extended to cover such areas as logic and other puzzles of deductive reasoning. ...

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 Arabian Nights. A palindromic prime is a prime number that is also a palindromic number. ... The On-Line Encyclopedia of Integer Sequences (OEIS) is a web-based searchable database of integer sequences. ... The term perfect square is used in mathematics in two meanings: a positive integer which is the square of some other integer, i. ... In the US postage stamp commemorating Buckminster Fuller and his contributions to architecture and science, some of his inventions are visible. ... This article needs to be wikified. ... For the story teller in the Arabian Nights or the 1001 Nights see Shahrazad. ... Jump to: navigation, search Queen Scheherazade tells her stories to King Shahryar. ...

Contents


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: Decimal, or less commonly, denary, usually refers to the base 10 numeral system. ... Natural number can mean either a positive integer (1, 2, 3, 4, ...) or a non-negative integer (0, 1, 2, 3, 4, ...). Natural numbers have two main purposes: they can be used for counting (there are 3 apples on the table), or they can be used for ordering (this is... Jump to: navigation, search A numeral is a symbol or group of symbols that represents a number. ... The radix (Latin for root), also called base, is the number of various unique symbols (or digits or numerals) a positional numeral system uses to represent numbers. ... In mathematics and computer science, a numerical digit is a symbol, e. ...

n=sum_{i=0}^ka_ib^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. Jump to: navigation, search 0 (zero), alternatively called naught or nought, is both a number and a numeral. ...


An alternative but equivalent definition is as follows. In an arbitrary but fixed base b, a number n is palindromic if and only if:

  • n consists of a single digit, or
  • n consists of two equal digits, or
  • n consists of three or more digits, the first and last digits are equal, and the number obtained by stripping the first and last digits off n is itself palindromic.

Decimal palindromic numbers

All numbers in base 10 with one digit {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} are palindromic ones. The number of palindromic numbers with two digits is 9: Jump to: navigation, search A numeral is a symbol or group of symbols that represents a number. ... Decimal, or less commonly, denary, usually refers to the base 10 numeral system. ... In mathematics and computer science, a numerical digit is a symbol, e. ... Jump to: navigation, search 0 (zero), alternatively called naught or nought, is both a number and a numeral. ... Jump to: navigation, search Look up one on Wiktionary, the free dictionary. ... 2 (two) is a number, numeral, and glyph. ... 1+1 ... 4 (four) is a number, numeral, and glyph. ... 5 (five) is a number, numeral, and glyph. ... 6 (six) is the natural number following 5 and preceding 7. ... Jump to: navigation, search 7 (seven) is the natural number following 6 and preceding 8. ... 8 (eight) is the natural number following 7 and preceding 9. ... Jump to: navigation, search 9 (nine) is the natural number following 8 and preceding 10. ...

{11, 22, 33, 44, 55, 66, 77, 88, 99}.

There are 90 palindromic numbers with three digits:

{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:

{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 OEIS). For some types of palindromic numbers these values are listed below in a table. Here 0 is included. The On-Line Encyclopedia of Integer Sequences (OEIS) is a web-based searchable database of integer sequences. ...

  101 102 103 104 105 106 107 108 109 1010
n natural 9 90 199 1099 1999 10999 19999 109999 199999
n even 5 9 49 89 489  + + + + +
n odd 5 10 60 110 610  + + + + +
n perfect square 3 6 13 14 19  + +
n prime 4 5 20 113 781 5953
n square-free 6 12 67 120 675  + + + + +
n non-square-free (μ(n)=0) 3 6 41 78 423  + + + + +
n square with prime root 2 3 5
n with an even number of distinct prime factors (μ(n)=1) 2 6 35 56 324 + + + + +
n with an odd number of distinct prime factors (μ(n)=-1) 5 7 33 65 352 + + + + +
n even with an odd number of prime factors                    
n even with an odd number of distinct prime factors 1 2 9 21 100 + + + + +
n odd with an odd number of prime factors 0 1 12 37 204 + + + + +
n odd with an odd number of distinct prime factors 0 0 4 24 139 + + + + +
n even squarefree with an even number of distinct prime factors 1 2 11 15 98 + + + + +
n odd squarefree with an even number of distinct prime factors 1 4 24 41 226 + + + + +
n odd with exactly 2 prime factors 1 4 25 39 205 + + + + +
n even with exactly 2 prime factors 2 3 11 64 + + + + +
n even with exactly 3 prime factors 1 3 14 24 122 + + + + +
n even with exactly 3 distinct prime factors                    
n odd with exactly 3 prime factors 0 1 12 34 173 + + + + +
n Carmichael number 0 0 0 0 0 1+ + + + +
n for which σ(n) is palindromic 6 10 47 114 688 + + + + +

Jump to: navigation, search Natural number can mean either a positive integer (1, 2, 3, 4, ...) or a non-negative integer (0, 1, 2, 3, 4, ...). Natural numbers have two main purposes: they can be used for counting (there are 3 apples on the table), and they can be used... In mathematics, the term even is used in several senses: an integer is called even if it is divisible by two; see even and odd numbers a function f defined on the real numbers is called even if f(x) = f(−x) for all x; see even and odd functions... In mathematics, any integer (whole number) is either even or odd. ... The term perfect square is used in mathematics in two meanings: a positive integer which is the square of some other integer, i. ... Jump to: navigation, search In mathematics, a prime number (or prime) is a natural number greater than one whose only positive divisors are one and itself. ... In mathematics, a square-free integer is one divisible by no perfect square, except 1. ... Jump to: navigation, search The classical Möbius function is an important multiplicative function in number theory and combinatorics. ... In number theory, the prime factors of a positive integer are the prime numbers that divide into that integer exactly, without leaving a remainder. ... In number theory, a Carmichael number is a composite positive integer n which satisfies the congruence bn − 1 ≡ 1 (mod n) for all integers b which are relatively prime to n (see modular arithmetic). ... In number theory, a multiplicative function is an arithmetic function f(n) of the positive integer n with the property that f(1) = 1 and whenever a and b are coprime, then f(ab) = f(a) f(b). ...

Other bases

Palindromic numbers can be considered in other numeral systems than decimal. For example, the binary palindromic numbers are: Jump to: navigation, search A numeral is a symbol or group of symbols that represents a number. ... Decimal, or less commonly, denary, usually refers to the base 10 numeral system. ... Look up binary in Wiktionary, the free dictionary Binary may mean: In mathematics and computer science, the binary (base-two) numeral system is a representation for numbers that uses only zeroes and ones as digits. ...

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, … (sequence A006995 in OEIS). The Mersenne primes form a subset of the binary palindromic primes. The On-Line Encyclopedia of Integer Sequences (OEIS) is a web-based searchable database of integer sequences. ... In mathematics, a Mersenne prime is a prime number that is one less than a power of two. ...


Generally, a number that is palindromic in one base is not palindromic in another base; for instance, 1646110 = 404D16. (The subscripts indicate radices, so n16 means n written in hexadecimal.) However, some numbers are copalindromic in several bases. The number 10510, for example, is palindromic in five bases: 12214 = 1518 = 7714 = 5520 = 3334. The year 1991 is palindromic in both decimal and hexadecimal (7C7). The radix (Latin for root), also called base, is the number of various unique symbols (or digits or numerals) a positional numeral system uses to represent numbers. ... In mathematics and computer science, hexadecimal, or simply hex, is a numeral system with a radix or base of 16 usually written using the symbols 0–9 and A–F or a–f. ...


In base 18, some powers of seven are palindromic:

 73 = 111 74 = 777 76 = 12321 79 = 1367631

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. A strictly non-palindromic number is an integer n that is not palindromic in any number base up to n − 2. ...


See also

A palindromic prime is a prime number that is also a palindromic number. ... Jump to: navigation, search A palindrome is a word, phrase, number or any other sequence of units (like a strand of DNA) which has the property of reading the same in either direction (the adjustment of spaces between letters is generally permitted). ...

External links


  Results from FactBites:
 
Palindromic prime - Wikipedia, the free encyclopedia (287 words)
A 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.
It's possible that a triply palindromic prime in base 10 may be also be palindromic in another base, such as base 2, but it would be highly remarkable if it was also a triply palindromic prime in that base as well.
Palindromic number - Wikipedia, the free encyclopedia (684 words)
The term palindromic is derived from palindrome, which refers to a word like rotor that remains unchanged under reversal of its letters.
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.
All numbers in base 10 with one digit {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} are palindromic ones.
  More results at FactBites »

 

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