Editing Palindromic number

{{Short description|Number that remains the same when its digits are reversed}}
A '''palindromic number''' (also known as a '''numeral palindrome''' or a '''numeric palindrome''') is a number (such as 16361) 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]], 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:
:  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, ... {{OEIS|id=A002113}}.

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:
* The [[palindromic prime]]s are 2, 3, 5, 7, 11, 101, 131, 151, ... {{OEIS|id=A002385}}.
* The palindromic [[square number]]s are 0, 1, 4, 9, 121, 484, 676, 10201, 12321, ... {{OEIS|id=A002779}}.

In any [[radix|base]] there are [[Infinite set|infinitely many]] palindromic numbers, since in any base the infinite [[sequence]] 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]] system, the concept of '''palindromicity''' can be applied to the [[natural numbers]] in any [[numeral system]]. Consider a number ''n''&nbsp;>&nbsp;0 in [[radix|base]] ''b''&nbsp;≥&nbsp;2, where it is written in standard notation with ''k''+1 [[numerical digit|digit]]s ''a''<sub>''i''</sub> as:
:<math>n=\sum_{i=0}^ka_ib^i</math>
with, as usual, 0&nbsp;≤&nbsp;''a''<sub>''i''</sub>&nbsp;<&nbsp;''b'' for all ''i'' and ''a''<sub>''k''</sub>&nbsp;≠&nbsp;0. Then ''n'' is palindromic if and only if ''a''<sub>''i''</sub>&nbsp;=&nbsp;''a''<sub>''k''&minus;''i''</sub> for all ''i''. [[0 (number)|Zero]] is written 0 in any base and is also palindromic by definition.

==Decimal palindromic numbers==
All numbers with one digit are palindromic, so in [[Decimal|base 10]] there are ten 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}.

All palindromic numbers with an even number of digits are divisible by [[11 (number)|11]].<ref>{{cite web |title=The Prime Glossary: palindromic prime |url=https://t5k.org/glossary/page.php?sort=PalindromicPrime |website=[[PrimePages]] |access-date=11 July 2023}}</ref>

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 smaller than 10<sup>4</sup>.

There are 1099 palindromic numbers smaller than 10<sup>5</sup> and for other exponents of 10<sup>n</sup> we have: 1999, 10999, 19999, 109999, 199999, 1099999, ... {{OEIS|id=A070199}}. The number of palindromic numbers which have some other property are listed below:
{| class="wikitable"
|-
! &nbsp;
! 10<sup>1</sup>
! 10<sup>2</sup>
! 10<sup>3</sup>
! 10<sup>4</sup>
! 10<sup>5</sup>
! 10<sup>6</sup>
! 10<sup>7</sup>
! 10<sup>8</sup>
! 10<sup>9</sup>
! 10<sup>10</sup>
|-
! style="font-weight:normal; text-align:left" | ''n'' [[Natural number|natural]]
| 10
| 19
| 109
| 199
| 1099
| 1999
| 10999
| 19999
| 109999
| 199999
|-
! style="font-weight:normal; text-align:left" | ''n'' [[even and odd numbers|even]]
| 5
| 9
| 49
| 89
| 489
| 889
| 4889
| 8889
| 48889
| 88889
|-
! style="font-weight:normal; text-align:left" | ''n'' [[odd number|odd]]
| 5
| 10
| 60
| 110
| 610
| 1110
| 6110
| 11110
| 61110
| 111110
|-
! style="font-weight:normal; text-align:left" | ''n'' [[square number|square]]
| colspan="2" | 4
| colspan="2" | 7
| 14
| 15
| colspan="2" | 20
| colspan="2" | 31
|-
! style="font-weight:normal; text-align:left" | ''n'' [[Cube (algebra)|cube]]
| colspan="2" | 3
| 4
| colspan="3" | 5
| colspan="3" | 7
| 8
|-
! style="font-weight:normal; text-align:left" | ''n'' [[prime number|prime]]
| 4
| 5
| colspan="2" | 20
| colspan="2" | 113
| colspan="2" | 781
| colspan="2" | 5953
|-
! style="font-weight:normal; text-align:left" | ''n'' [[square-free integer|squarefree]]
| 6
| 12
| 67
| 120
| 675
| 1200
| 6821
| 12160
| +
| +
|-
! style="font-weight:normal; text-align:left" | ''n'' non-squarefree ([[Möbius function|μ(''n'')]]=0)
| 4
| 7
| 42
| 79
| 424
| 799
| 4178
| 7839
| +
| +
|-
! style="font-weight:normal; text-align:left" | ''n'' square with prime root<ref>{{OEIS|A065379}} The next example is 19 digits - 900075181570009.</ref>
| colspan="1" | 2
| colspan="2" | 3
| colspan="6" | 5
|-
! style="font-weight:normal; text-align:left" | ''n'' with an even number of distinct [[prime factor]]s (μ(''n'')=1)
| 2
| 6
| 35
| 56
| 324
| 583
| 3383
| 6093
| +
| +
|-
! style="font-weight:normal; text-align:left" | ''n'' with an odd number of distinct prime factors (μ(''n'')=-1)
| 4
| 6
| 32
| 64
| 351
| 617
| 3438
| 6067
| +
| +
|-
! style="font-weight:normal; text-align:left" | ''n'' even with an odd number of prime factors
| 1
| 2
| 9
| 21
| 100
| 180
| 1010
| 6067
| +
| +
|-
! style="font-weight:normal; text-align:left" | ''n'' even with an odd number of distinct prime factors
| 3
| 4
| 21
| 49
| 268
| 482
| 2486
| 4452
| +
| +
|-
! style="font-weight:normal; text-align:left" | ''n'' odd with an odd number of prime factors
| 3
| 4
| 23
| 43
| 251
| 437
| 2428
| 4315
| +
| +
|-
! style="font-weight:normal; text-align:left" | ''n'' odd with an odd number of distinct prime factors
| 4
| 5
| 28
| 56
| 317
| 566
| 3070
| 5607
| +
| +
|-
! style="font-weight:normal; text-align:left" | ''n'' even squarefree with an even number of (distinct) prime factors
| 1
| 2
| 11
| 15
| 98
| 171
| 991
| 1782
| +
| +
|-
! style="font-weight:normal; text-align:left" | ''n'' odd squarefree with an even number of (distinct) prime factors
| 1
| 4
| 24
| 41
| 226
| 412
| 2392
| 4221
| +
| +
|-
! style="font-weight:normal; text-align:left" | ''n'' odd with exactly 2 prime factors
| 1
| 4
| 25
| 39
| 205
| 303
| 1768
| 2403
| +
| +
|-
! style="font-weight:normal; text-align:left" | ''n'' even with exactly 2 prime factors
| 2
| 3
| colspan="2" | 11
| colspan="2" | 64
| colspan="2" | 413
| +
| +
|-
! style="font-weight:normal; text-align:left" | ''n'' even with exactly 3 prime factors
| 1
| 3
| 14
| 24
| 122
| 179
| 1056
| 1400
| +
| +
|-
! style="font-weight:normal; text-align:left" | ''n'' even with exactly 3 distinct prime factors
| 0
| 1
| 18
| 44
| 250
| 390
| 2001
| 2814
| +
| +
|-
! style="font-weight:normal; text-align:left" | ''n'' odd with exactly 3 prime factors
| 0
| 1
| 12
| 34
| 173
| 348
| 1762
| 3292
| +
| +
|-
! style="font-weight:normal; text-align:left" | ''n'' [[Carmichael number]]
| 0
| 0
| 0
| 0
| 0
| 1
| 1
| 1
| 1
| 1
|-
! style="font-weight:normal; text-align:left" | ''n'' for which [[Divisor function|σ(''n'')]] is palindromic
| 6
| 10
| 47
| 114
| 688
| 1417
| 5683
| +
| +
| +
|}

===Perfect powers===
There are many palindromic [[perfect power]]s ''n''<sup>''k''</sup>, where ''n'' is a natural number and ''k'' is 2, 3 or 4.
* Palindromic [[Square number|squares]]: 0, 1, 4, 9, 121, 484, 676, 10201, 12321, 14641, 40804, 44944, ... {{OEIS|id=A002779}}
* Palindromic [[Cube (algebra)|cubes]]: 0, 1, 8, 343, 1331, 1030301, 1367631, 1003003001, ... {{OEIS|id=A002781}}
* Palindromic [[fourth power]]s: 0, 1, 14641, 104060401, 1004006004001, ... {{OEIS|id=A186080}}

The first nine terms of the sequence 1<sup>2</sup>, 11<sup>2</sup>, 111<sup>2</sup>, 1111<sup>2</sup>, ... form the palindromes 1, 121, 12321, 1234321, ... {{OEIS|id=A002477}}

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 (10<sup>n</sup> + 1).

[[Gustavus Simmons]] conjectured there are no palindromes of form ''n''<sup>''k''</sup> for ''k'' > 4 (and ''n'' > 1).<ref>Murray S. Klamkin (1990), ''Problems in applied mathematics: selections from SIAM review'', [https://books.google.com/books?id=WI9ZGl3M8bYC&pg=PA520 p. 520].</ref>

==Other bases==
Palindromic numbers can be considered in [[numeral system]]s other than [[decimal]]. For example, the [[Binary numeral system|binary]] palindromic numbers are those with the binary representations:
:0, 1, 11, 101, 111, 1001, 1111, 10001, 10101, 11011, 11111, 100001, ... {{OEIS|A057148}}
or in decimal:
:0, 1, 3, 5, 7, 9, 15, 17, 21, 27, 31, 33, ... {{OEIS|A006995}}
The [[Fermat prime]]s and the [[Mersenne prime]]s form a subset of the binary palindromic primes.

Any number <math>n</math> is palindromic in all bases <math>b</math> with <math>b > n</math> (trivially so, because <math>n</math> is then a single-digit number), and also in base <math>n-1</math> (because <math>n</math> is then <math>11_{n-1}</math>). Even excluding cases where the number is smaller than the base, most numbers are palindromic in more than one base. For example, <math>1221_4=151_8=77_{14}=55_{20}=33_{34}=11_{104}</math>, <math>1991_{10}=7C7_{16}</math>. A number <math>n</math> is never palindromic in base <math>b</math> if <math>n/2 \le b \le n-2</math>. Moreover, a prime number <math>p</math> is never palindromic in base <math>b</math> if <math>\sqrt{p} < b < p-1</math>.

A number that is non-palindromic in all bases ''b'' in the range 2&nbsp;≤&nbsp;''b''&nbsp;≤&nbsp;''n''&nbsp;&minus;&nbsp;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 <math>n > 6</math> is composite, then either <math>n = ab</math> for some <math>1 < a < b-1</math>, in which case ''n'' is the palindrome "aa" in base <math>b-1</math>, or else it is a perfect square <math>n = a^2</math>, in which case ''n'' is the palindrome "121" in base <math>a-1</math> (except for the special case of <math>n = 9 = 1001_2</math>).<ref>{{Cite OEIS|A016038|Strictly non-palindromic numbers}}</ref><ref>{{Cite journal|last1=Guy|first1=Richard K.|author-link=Richard Guy|date=1989|title=Conway's RATS and other reversals|journal=The American Mathematical Monthly|volume=96|number=5|pages=425–428|doi=10.2307/2325149 |jstor=2325149}}</ref>

The first few strictly non-palindromic numbers {{OEIS|id=A016038}} are:

:[[0 (number)|0]], [[1 (number)|1]], [[2 (number)|2]], [[3 (number)|3]], [[4 (number)|4]], [[6 (number)|6]], [[11 (number)|11]], [[19 (number)|19]], [[47 (number)|47]], [[53 (number)|53]], [[79 (number)|79]], [[103 (number)|103]], [[137 (number)|137]], [[139 (number)|139]], [[149 (number)|149]], [[163 (number)|163]], [[167 (number)|167]], [[179 (number)|179]], [[223 (number)|223]], [[263 (number)|263]], [[269 (number)|269]], [[283 (number)|283]], [[293 (number)|293]], 311, 317, 347, 359, 367, 389, 439, 491, 563, 569, 593, 607, 659, 739, 827, 853, 877, 977, 983, 997, ...

==Antipalindromic numbers==
If the digits of a natural number don't only have to be reversed in order, but also subtracted from <math>b-1</math> 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 <math>a_i</math> in base <math>b</math>, a number is antipalindromic [[iff]] <math>a_i = b - 1 - a_{k-i}</math>.<ref>{{cite arXiv |last1=Dvorakova |first1=Lubomira  |last2=Kruml |first2=Stanislav |last3=Ryzak |first3=David |eprint=2008.06864 |title=Antipalindromic numbers |class= math.CO|date=16 Aug 2020 }}</ref>

==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 [[oeis:A281509|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 [[oeis:A281508|A281508]].

==Sum of the reciprocals==
The sum of the reciprocals of the palindromic numbers is a convergent series, whose value is approximately 3.37028... {{OEIS|id=A118031}}.

==Scheherazade numbers==
'''Scheherazade numbers''' are a set of numbers identified by [[Buckminster Fuller]] in his book ''Synergetics''.<ref>R. Buckminster Fuller, with E. J. Applewhite, [http://www.rwgrayprojects.com/synergetics/s12/p2200.html#1230.00 ''Synergetics: Explorations in the Geometry of thinking''] {{Webarchive|url=https://web.archive.org/web/20160227163051/http://www.rwgrayprojects.com/synergetics/s12/p2200.html#1230.00 |date=2016-02-27 }}, Macmillan, 1982 {{ISBN|0-02-065320-4}}.</ref>  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 coefficient]]s. For instance,
:<math>(1001)^6 = 1,006,015,020,015,006,001 </math>

This sequence fails at (1001)<sup>13</sup> because there is a [[Carry (arithmetic)|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.<ref>Fuller, [http://www.rwgrayprojects.com/synergetics/s12/p3100.html pp. 773-774] {{Webarchive|url=https://web.archive.org/web/20160305202829/http://www.rwgrayprojects.com/synergetics/s12/p3100.html |date=2016-03-05 }}</ref>  Many other Scheherazade numbers show similar symmetries when expressed in this way.<ref>Fuller, pp. 777-780</ref>

== 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.<ref>{{Cite journal|last1=Cilleruelo|first1=Javier|last2=Luca|first2=Florian|last3=Baxter|first3=Lewis|date=2016-02-19|title=Every positive integer is a sum of three palindromes|url=https://www.ams.org/journals/mcom/2018-87-314/S0025-5718-2017-03221-X/home.html|journal=Mathematics of Computation|arxiv=1602.06208|access-date=2021-04-28|archive-date=2021-02-12|archive-url=https://web.archive.org/web/20210212212120/https://www.ams.org/journals/mcom/2018-87-314/S0025-5718-2017-03221-X/home.html|url-status=live}} ([https://arxiv.org/abs/1602.06208 arXiv preprint] {{Webarchive|url=https://web.archive.org/web/20190208100101/https://arxiv.org/abs/1602.06208 |date=2019-02-08 }})</ref>

==Notes==
{{Reflist}}

==References==
*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 ([https://books.google.com/books?id=Am9og6q_ny4C&dq=palindromic+number&pg=PT69 Limited Online-Version (Google Books)])

==External links==
*{{MathWorld|urlname=PalindromicNumber|title= Palindromic Number}}
*[http://www.jasondoucette.com/worldrecords.html Jason Doucette - 196 Palindrome Quest / Most Delayed Palindromic Number]
*[https://web.archive.org/web/20061104023524/http://www.p196.org/ 196 and Other Lychrel Numbers]
*[http://www.mathpages.com/home/kmath359.htm On General Palindromic Numbers] at MathPages
*[http://mathforum.org/library/drmath/view/57170.html Palindromic Numbers to 100,000] from Ask Dr. Math
*[http://users.skynet.be/worldofnumbers/cube.htm P. De Geest, Palindromic cubes]
*[[Yutaka Nishiyama]], [http://ijpam.eu/contents/2012-80-3/9/9.pdf Numerical Palindromes and the 196 Problem], IJPAM, Vol.80, No.3, 375–384, 2012.
{{Classes of natural numbers}}

[[Category:Base-dependent integer sequences]]
[[Category:Palindromes]]

[[pl:Palindrom#Palindromy liczbowe]]