Palindromic number

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. 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 primes are 2, 3, 5, 7, 11, 101, 131, 151, … (sequence A002385 in OEIS)
• the palindromic perfect squares are 0, 1, 4, 9, 121, 484, 676, 10201, 12321, …

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.

Contents 1 Formal definition 2 Decimal palindromic numbers 3 Other bases 4 Lychrel process 5 Colloquial terms in other languages 6 See also 7 External links

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 a_i as:

<math>n=\sum_{i=0}^ka_ib^i</math>

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. 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:

{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 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	+	+	+	+	+

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) and last year (2007) is also palindromic (7D7). In base 18, some powers of seven are palindromic:

73 =     111
74 =     777
76 =   12321
79 = 1367631
E3 =     888

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 11_n−1). A number that is non-palindromic in all bases 2 ≤ b < n − 1 is called a strictly non-palindromic number.

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.

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 capicua, origined by the two Catalan words "cap", that means head, and "cua", 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.

See also

• Lychrel number
• Palindromic prime

External links

• Jason Doucette - 196 Palindrome Quest / Most Delayed Palindromic Number
• 196 and Other Lychrel Numbers
• Palindromic Numbers to 100,000 from Ask Dr. Math
• 111111111x111111111.com