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:

:$n=\backslash sum\_\{i=0\}^ka\_ib^i$

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 is written 0 in any base and is also palindromic by definition.

All numbers with one digit are palindromic, so in 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.{{cite web |title=The Prime Glossary: palindromic prime |url=https://t5k.org/glossary/page.php?sort=PalindromicPrime |website=PrimePages |access-date=11 July 2023}}

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 10⁴.

There are 1099 palindromic numbers smaller than 10⁵ and for other exponents of 10ⁿ we have: 1999, 10999, 19999, 109999, 199999, 1099999, ... {{OEIS|id=A070199}}. The number of palindromic numbers which have some other property are listed below:

There are many palindromic perfect powers n^k, where n is a natural number and k is 2, 3 or 4.

• Palindromic squares: 0, 1, 4, 9, 121, 484, 676, 10201, 12321, 14641, 40804, 44944, ... {{OEIS|id=A002779}}
• Palindromic cubes: 0, 1, 8, 343, 1331, 1030301, 1367631, 1003003001, ... {{OEIS|id=A002781}}
• Palindromic fourth powers: 0, 1, 14641, 104060401, 1004006004001, ... {{OEIS|id=A186080}}

The first nine terms of the sequence 1², 11², 111², 1111², ... 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 (10ⁿ + 1).

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

Palindromic numbers can be considered in numeral systems other than decimal. For example, the 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 primes and the Mersenne primes 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 $11\_\{n-1\}$). Even excluding cases where the number is smaller than the base, most numbers are palindromic in more than one base. For example, $1221\_4=151\_8=77\_\{14\}=55\_\{20\}=33\_\{34\}=11\_\{104\}$, $1991\_\{10\}=7C7\_\{16\}$. A number $n$ is never palindromic in base $b$ if $n/2\; \backslash le\; b\; \backslash le\; n-2$. Moreover, a prime number $p$ is never palindromic in base $b$ if .

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 , 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 $n\; =\; 9\; =\; 1001\_2$).{{Cite OEIS|A016038|Strictly non-palindromic numbers}}{{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}}

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

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

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 $a\_i\; =\; b\; -\; 1\; -\; a\_\{k-i\}$.{{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 }}

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... {{OEIS|id=A118031}}.

Scheherazade numbers are a set of numbers identified by Buckminster Fuller in his book Synergetics.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}}. 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,

:$(1001)^6\; =\; 1,006,015,020,015,006,001$

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.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 }} Many other Scheherazade numbers show similar symmetries when expressed in this way.Fuller, pp. 777-780

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.{{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 }})

{{Reflist}}

• 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)])

• {{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