Palindromic number

A palindromic number (also known as a numeral palindrome or a numeric palindrome) is a number that remains the same when its digits are reversed. Like 16461, for example, it is "symmetrical". 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, … (sequence A002113 in the OEIS).

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 identified a set of numbers he called Scheherazade numbers, some of which have a palindromic symmetry of digit groups.

It is fairly straightforward to appreciate (and prove) 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.

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:

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.

Decimal palindromic numbers

Decimal palindromic numbers with an even number of digits are divisible by 11.

All numbers in base 10 (and indeed in any base) with one digit 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: 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, … (sequence A070199 in the OEIS). 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 10 19 109 199 1099 1999 10999 19999 109999 199999
n even 5 9 49 89 489 889 4889 8889 48889 88889
n odd 5 10 60 110 610 1110 6110 11110 61110 111110
n square 4 7 14 15 20 31
n cube 3 4 5 7 8
n prime 4 5 20 113 781 5953
n squarefree 6 12 67 120 675 1200 6821 12160 + +
n non-squarefree (μ(n)=0) 4 7 42 79 424 799 4178 7839 + +
n square with prime root[1] 2 3 5
n with an even number of distinct prime factors (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) 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 squares: 0, 1, 4, 9, 121, 484, 676, 10201, 12321, 14641, 40804, 44944, ... (sequence A002779 in the OEIS)
  • Palindromic cubes: 0, 1, 8, 343, 1331, 1030301, 1367631, 1003003001, ... (sequence A002781 in the OEIS)
  • Palindromic fourth powers: 0, 1, 14641, 104060401, 1004006004001, ... (sequence A186080 in the OEIS)

The first nine terms of the sequence 12, 112, 1112, 11112, ... form the palindromes 1, 121, 12321, 1234321, ... (sequence A002477 in the OEIS)

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 nk for k > 4 (and n > 1).[2]

Other bases

Palindromic numbers can be considered in numeral systems other than decimal. For example, the binary 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, … (sequence A006995 in the OEIS). The Fermat primes and the Mersenne primes 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, for example, ,

In base 7, because 1017 is twice a perfect square (52=347), several of its multiples are palindromic squares:

132=202
262=1111
552=4444
1012=10201
1432=24442

In base 18, some powers of seven are palindromic:

70=1
71=7
73=111
74=777
76=12321
79=1367631

And in base 24 the first eight powers of five are palindromic as well:

50=1
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.

A palindromic number in base b that is made up of palindromic sequences of length l arranged in a palindromic order (such as 101 111 010 111 1012) is palindromic in base bl (for example the above binary number is palindromic in base 23=8 (it is equal to 572758))

The square of 13310 in base 30 is 4D302 = KKK30 = 3R362 = DPD36. In base 24 there are more palindromic squares due to 52 = 11. And squares of all numbers in the form 16...7 (with any number of 6'es between the 1 and 7) are palindromic. 1672 = 1E5E1, 16672 = 1E3K3E1, 166672 = 1E3H8H3E1.

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.

For example, 1,186,060,307,891,929,990 takes 261 iterations to reach the 119-digit palindrome 44562665878976437622437848976653870388884783662598425855963436955852489526638748888307835667984873422673467987856626544, which is the current world record for the Most Delayed Palindromic Number. It was solved by Jason Doucette's algorithm and program (using Benjamin Despres' reversal-addition code) on November 30, 2005.

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... (sequence A118031 in the OEIS).

Scheherazade numbers

Scheherazade numbers are a set of numbers identified by Buckminster Fuller in his book Synergetics.[3] 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,

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.[4] Many other Scheherazade numbers show similar symmetries when expressed in this way.[5]

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.[6]

gollark: Nope! You could only prove i am isomorphic to them, and in any case I have countermeasures against being put in sets.
gollark: They ask very politely.
gollark: ↑ you, then
gollark: https://media.discordapp.net/attachments/392020612908056576/819104214759637002/tumblr_mg4pppjhiP1rxn93po1_500.png
gollark: If it's not good, andrew, I WILL deploy a modified version of this proof.

See also

Notes
  1. (sequence A065379 in the OEIS) The next example is 19 digits - 900075181570009.
  2. Murray S. Klamkin (1990), Problems in applied mathematics: selections from SIAM review, p. 520.
  3. R. Buckminster Fuller, with E. J. Applewhite, Synergetics: Explorations in the Geometry of thinking, Macmillan, 1982 ISBN 0-02-065320-4.
  4. Fuller, pp. 773-774
  5. Fuller, pp. 777-780
  6. Cilleruelo, Javier; Luca, Florian; Baxter, Lewis (2016-02-19). "Every positive integer is a sum of three palindromes". Mathematics of Computation. (arXiv preprint)

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 (Limited Online-Version (Google Books))
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