Categories of Palindromic Numbers
|No. of Digits||Prime||Triangular||Perfect square||sum of consecutive squares (2 or more)||cube||product of successive numbers|
Properties of Palindromic Numbers
|*Palindromes have their origin from the Greek word palindromos meaning "running back again".|
|* Take any number. Add it to the reverse number. If it it is not a palindrome, again add the new number to its reverse till it becomes a palindrome. 80% of the numbers under 10000 yield palindromes in 4 or less steps. about 10% in more than 4 but up to 7 steps. A rare number 89 as well as 98 take 24 iterations to become a palindrome. Palindrome words -- eye, noon, radar, mom, dad, deed, civic, level|
|* There are some numbers which do not form a palindrome in the above process. These are called Lychrels. The first Lychrels is 196. The reverse no. 691 is also Lychrels. If one goes on adding these numbers as above, all the numbers generated are lychrels. For example, in the chain of (196,691)-->(788,887), (1675, 5761) and up above are all lychrels.|
|* All palindromes with even number of digits are divisible by 11.Therefore barring 11 which is a 2 digit Palindrome, there are no palindromes with 4,6, 8, ..... no. of digits. All even digit palindromes can be expressed as 100a( 101 + 1) + 10b (103 + 1) + c (105 +1) +.... ( this is the format for 6 digit palindrome. One can construct on similar lines for more or less number of even digits ) & each co-efficient reduces to the equation 10 (2k+1) +1 = 11x where k, x have integer values.|
|* All Palindromic numbers which are triangular must have either of 0,1,3,5,6,8 as their last digit and the lower of the product digits of the triangular number has to be even. In other words for n(n+1)/2 to be palindromic, n is to an even number.|
* In molecular biology, the DNA and RNA sequences that read the same from both the ends i.e 5' & 3' end are called Palindromes. The sites of many restriction enzymes such as Restriction Endonuclease that cut DNA are palindromes.
Long DNA palindromes pose a threat to genome stability. This instability is primarily mediated by slippage on the lagging strand of the replication fork between short directly repeated sequences close to the ends of the palindrome. The role of the palindrome is likely to be the juxtaposition of the directly repeated sequences by intra-strand base-pairing. This intra-strand base-pairing, if present on both strands, results in a cruciform structure. In bacteria, cruciform structures have proved difficult to detect in vivo, suggesting that if they form, they are either not replicated or are destroyed. SbcCD, a recently discovered exonuclease of Escherichia coli, is responsible for preventing the replication of long palindromes. These observations lead to the proposal that cells may have evolved a post-replicative mechanism for the elimination and/or repair of large DNA secondary structures
It has been found that Direct and inverted repeats elicit genetic instability by both exploiting and eluding DNA double-strand break repair systems in mycobacteria.
It has also been found that widespread and non-random distribution of DNA palindromes in cancer cells provides a structural platform for subsequent gene amplification
|* Frequency of Palindrome distribution :
(n-1) k/2 where k is even and
(n-1) (k+1)/2 where k is odd.
n represents the Base system and k is the total number of digits in the
* In general for a number in any base n with k digits the number
of unique reverses of k digits is given by
* General formulae which give the frequency of palindromes for all numbers in any base n less than or equal to k digits.
If k is even, the formula = 2(n - l)[(n - 1)k/2 - 1]
If k is odd, formula = 2(n - l)[(n - 1)(k+1)/2 - 1]
- (n - 1)(k+1)/2
References -- Palindromic Numbers