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palindromic number





A number such as 1234321 that reads the same forwards and backwards; more generally, a symmetrical number written in some base a as a1 a2 a3 ...|... a3 a2 a1. In the familiar base 10 system, there are nine two-digit palindromic numbers: 11, 22, 33, 44, 55, 66, 77, 88, 99; there are 90 palindromics with three digits: 101, 111, 121, 131, 141, 151, 161, 171, 181, 191, ..., 909, 919, 929, 939, 949, 959, 969, 979, 989, 999; and there are 90 palindromics with four digits: 1001, 1111, 1221, 1331, 1441, 1551, 1661, 1771, 1881, 1991, ..., 9009, 9119, 9229, 9339, 9449, 9559, 9669, 9779, 9889, 9999, giving a total of 199 palindromic numbers below 104. Below 105 there are 1099 palindromics and for other exponents of 10n there are 1999, 10999, 19999, 109999, 199999, 1099999, ... It is conjectured, but has not been proved, that there are an infinite number of palindromic prime numbers.

With the exception of 11, palindromic primes must have an odd number of digits. A normally quick way to produce a palindromic number is to pick a positive integer of two or more digits, reverse the digits, and add to the original, then repeat this process with the new number, and so on. For example, 3462 gives the sequence 3462, 6105, 11121, 23232. Does the series formed by adding a number to its reverse always end in a palindrome? It used to be thought so. However, this conjecture has been proved false for bases 2, 4, 8, and other powers of 2, and seems to be false for base 10 as well. Among the first 100,000 numbers, 5,996 numbers are known that have not produced palindromic numbers by the add-and-reverse method in calculations carried out to date. The first few of these are 196, 887, 1675, 7436, 13783, 52514, ... A proof that these numbers never produce palindromes, however, has yet to be found. The largest known palindromic prime, containing 30,913 digits was found by David Broadhurst in 2003.


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   • TYPES OF NUMBERS