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. The first palindromic numbers 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, … (sequence A002113 in 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 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.

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[edit] 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:

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.

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.

[edit] Decimal palindromic numbers

All numbers in base 10 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:

{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.

  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 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 + + + + +
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 + + +
n even with exactly 3 distinct prime factors 0 1 18 44 250 390 2001 + + +
n odd with exactly 3 prime factors 0 1 12 34 173 348 1762 + + +
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 + + +

[edit] Other bases

Palindromic numbers can be considered in other numeral systems 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 OEIS). The Mersenne primes form a subset of the binary palindromic primes.

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 number 1991 is palindromic in both decimal and hexadecimal (7C7).

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.

[edit] Lychrel process

Non-palindromic numbers can be paired with palendromic 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.

[edit] Colloquial terms in other languages

  • In Catalonia, and through Catalan influence in Spain and by extension in the Spanish-speaking countries and Portugal (through Spanish), exists the specific word capicúa, origined by the two Catalan words "cap", that means head, and "cúa", that means tail. The conjunction "i" (and) joins the three words in a new one that explains in itself the symmetry. This word is widely used in the colloquial language, rather than palindromic number (mathematics) or palindrome, for words, phrases or others.

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