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Palindromic Number

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1 reference results for: Palindromic number

Wikipedia

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 (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, … .

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, … ()
the palindromic square numbers 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.

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:

n=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.

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 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, … . 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	+	+	+

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, … . 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

5^A =     15AA51

5^C =    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 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.

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 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.

References

Malcolm E. Lines: A Number for Your Thoughts: Facts and Speculations about Number from Euclid to the latest Computers: CRC Press 1986, ISBN 0852744951, S. 61 (Limited Online-Version (Google Books))