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# Palindromic numbers

A base-${\displaystyle b}$ palindromic number is a positive integer that reads the same whether its digits are read forward or backward, i.e. it is a palindrome in base ${\displaystyle b}$. For example, 93 is palindromic in binary: 101101. When ${\displaystyle b}$ is left unspecified, it is generally assumed to be 10.

Almost all palindromic numbers are composite, few are palindromic primes.

We may regard base ${\displaystyle \scriptstyle b\,}$ palindromic numbers as belonging to one of four categories:

• {0}, and
• the base ${\displaystyle \scriptstyle b\,}$ nonzero 1-digit numbers (there are ${\displaystyle \scriptstyle b-1\,}$ of these), and
• the base ${\displaystyle \scriptstyle b\,}$ 2-digits (repdigits) numbers (there are also ${\displaystyle \scriptstyle b-1\,}$ of these), and
• starting from ${\displaystyle b^{2}+1}$ (the first 3 digits palindrome in the given base ${\displaystyle b}$).

Obviously, all repdigits, which include repunits, are palindromic.

The following table lists for a few bases a few palindromic numbers in the latter two categories mentioned above.

Base ${\displaystyle b}$ palindromic numbers
${\displaystyle \scriptstyle b\,}$ Sequences A-number
2 {3} & {5, 7, 9, 15, 17, 21, 27, 31, 33, 45, 51, 63, 65, 73, 85, 93, 99, 107, 119, 127, 129, 153, 165, 189, 195, 219, 231, 255, 257, 273, 297, 313, 325, 341, 365, 381, 387, 403, ...} A006995
3 {4, 8} & {10, 13, 16, 20, 23, 26, 28, 40, 52, 56, 68, 80, 82, 91, 100, 112, 121, 130, 142, 151, 160, 164, 173, 182, 194, 203, 212, 224, 233, 242, 244, 280, 316, 328, 364, 400, ...} A014190
4 {5, 10, 15} & {17, 21, 25, 29, 34, 38, 42, 46, 51, 55, 59, 63, 65, 85, 105, 125, 130, 150, 170, 190, 195, 215, 235, 255, 257, 273, 289, 305, 325, 341, 357, 373, 393, 409, ...} A014192
5 {6, 12, 18, 24} & {26, 31, 36, 41, 46, 52, 57, 62, 67, 72, 78, 83, 88, 93, 98, 104, 109, 114, 119, 124, 126, 156, 186, 216, 246, 252, 282, 312, 342, 372, 378, 408, 438, ...} A029952
6 {7, 14, 21, 28, 35} & {37, 43, 49, 55, 61, 67, 74, 80, 86, 92, 98, 104, 111, 117, 123, 129, 135, 141, 148, 154, 160, 166, 172, 178, 185, 191, 197, 203, 209, 215, 217, ...} A029953
7 {8, 16, 24, 32, 40, 48} & {50, 57, 64, 71, 78, 85, 92, 100, 107, 114, 121, 128, 135, 142, 150, 157, 164, 171, 178, 185, 192, 200, 207, 214, 221, 228, 235, 242, 250, ...} A029954
8 {9, 18, 27, 36, 45, 54, 63} & {65, 73, 81, 89, 97, 105, 113, 121, 130, 138, 146, 154, 162, 170, 178, 186, 195, 203, 211, 219, 227, 235, 243, 251, 260, 268, 276, ...} A029803
9 {10, 20, 30, 40, 50, 60, 70, 80} & {82, 91, 100, 109, 118, 127, 136, 145, 154, 164, 173, 182, 191, 200, 209, 218, 227, 236, 246, 255, 264, 273, 282, 291, 300, ...} A029955
10 {11, 22, 33, 44, 55, 66, 77, 88, 99} & {101, 111, 121, 131, 141, 151, 161, 171, 181, 191, 202, 212, 222, 232, 242, 252, 262, 272, 282, 292, 303, 313, 323, 333, ...} A002113

In Roman numerals, most palindromic numbers are trivially palindromic due to either having just one "digit" or being repdigits, though 19 (as XIX), 190 (as CXC) and 1900 (as MCM) are exceptions.

A078715 Palindromic Roman numerals.

{1, 2, 3, 5, 10, 19, 20, 30, 50, 100, 190, 200, 300, 500, 1000, 1900, 2000, 3000}

represented as Roman numerals

{I, II, III, V, X, XIX, XX, XXX, L, C, CXC, CC, CCC, D, M, MCM, MM, MMM}

The date September 11 rendered in Roman numerals is the palindrome "IX XI" sometimes used on counterterrorism unit patches.

## Iterated reverse digits and add

In the iterated reverse digits and add process, most starting values reach a palindrome sooner or later.

A061563 Start with ${\displaystyle n}$; add to itself with digits reversed; if palindrome, stop; otherwise repeat; ${\displaystyle a(n)}$ gives palindrome at which it stops, or -1 if [apparently, not yet proved] no palindrome is ever reached.

{0, 2, 4, 6, 8, 11, 33, 55, 77, 99, 11, 22, 33, 44, 55, 66, 77, 88, 99, 121, 22, 33, 44, 55, 66, 77, 88, 99, 121, 121, 33, 44, 55, 66, 77, 88, 99, 121, 121, 363, 44, 55, 66, 77, 88, ...}

A033665 Number of 'Reverse and Add' steps needed to reach a palindrome, or -1 if [apparently, not yet proved] never reaches a palindrome.

{0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 0, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 0, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 0, 1, 2, 1, 2, 2, 1, 1, 1, 1, 1, 0, 1, 2, 2, 3, ...}

89, for example, reaches a palindrome in 24 steps.

A033670 Reverse and Add! trajectory of 89. (${\displaystyle a(0)}$ = 89, ${\displaystyle a(24)}$ = 8813200023188, a palindrome)

{89, 187, 968, 1837, 9218, 17347, 91718, 173437, 907808, 1716517, 8872688, 17735476, 85189247, 159487405, 664272356, 1317544822, 3602001953, 7193004016, 13297007933, 47267087164, 93445163438, 176881317877, 955594506548, 1801200002107, 8813200023188}

It's not known if 196 ever does; A006960 lists the first 2390 values of its reverse and add sequence.

A006960 Reverse and Add! sequence starting with 196.

{196, 887, 1675, 7436, 13783, 52514, 94039, 187088, 1067869, 10755470, 18211171, 35322452, 60744805, 111589511, 227574622, 454050344, 897100798, 1794102596, 8746117567, 16403234045, ...}

196 is conjectured to be smallest initial term which does not lead to a palindrome. John Walker, Tim Irvin and others have extended the trajectory of 196 to millions of digits without finding a palindrome.