Prime number palindrome

See also: Palindrome (homonymy)

In Mathematical, a prime number palindrome is a Prime number which is also a Nombre palindrome. The character palindrome depends on the bases numbering system and its conventions of writing, while the primality is independent of this kind of considerations.

The first prime numbers palindromes in bases 10 are:

2, 3, 5, 7, 11, 101 131, 151, 181, 191, 313,353,373,383,727,757,787,797,919,929,10301,10501,10601,11311,11411,12421,12721,12821,13331,13831,13931,14341,14741,15451,15551,16061,16361,16561,16661,17471,17971,18181,18481,19391,19891,19991

It can be noted above in list, that there do not exist prime numbers palindromes with 2 or 4 digits, except for 11. If one considers the test of Divisibilité for 11, it can be deduced that any palindrome number with an even number of figures is divisible by 11.

One is unaware of if there exists an infinity of prime numbers palindromes bases 10 of them. The greatest prime number palindrome known is $10^\; \{130022\}\; +\; 3761673\; \backslash \; times\; 10^\; \{65008\}\; +\; 1\; \backslash ,$, which was discovered by Harvey Dubner the November 7th 2004 and was announced the November 19th 2004.

Into binary, the prime numbers palindromes easiest to obtain are the prime numbers of Mersenne, since they are also prime numbers uniforms. The first four prime numbers palindromes (which are not numbers of Mersenne) are 5 (101), 17 (10001), 73 (1001001) and 107 (1101011).

Ribenboim defines a prime number the tripling palindrome like a number which, in addition to being a prime number palindrome, has a number of figures which is itself a prime number palindrome. For example, $10^\; \{11310\}\; +\; 4661664\; \backslash \; times\; 10^\; \{5652\}\; +\; 1\; \backslash ,$, has 11311 digits. It is possible that a prime number the tripling palindrome bases 10 of them can be also palindrome in another base, such as base 2, but it would be highly remarkable if it were also three times over palindrome in this base.

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