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Old 2003-04-12, 17:55   #1
S80780
 
Jan 2003
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Default Palindromic Primes

Hi, all!

When dealing with primes, you often hear about twin primes and the twin prime conjecture. I'd like to know if there is a similar conjecture or even theorem dealing with palindromic primes like e.g. 11, 101, and twin palindromic primes like 13 and 31, 17 and 71 etc. Would it be base-independent?

Benjamin

Some palindrome stuff
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Old 2003-04-13, 02:41   #2
trif
 
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Well, 13 and 31 are 1101 and 11111 in base 2 respectively, so it would certainly be base dependent.
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Old 2003-04-13, 08:34   #3
S80780
 
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Not certainly. What I mean is the infinite amount of such primes. Their values would, of course, vary base dependent.
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Old 2003-04-13, 17:25   #4
jocelynl
 
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Hi Benjamin,

How about base 2.

11 and 13, 23 and 29, 47 and 61, 191 and 253. And non prime 95 and 125.
There seems to be one in base 2.

formula is x(0)=5
x(n) = x(n-1)*2+1

y(0)=5
y(n)=y(n-1)*2+3

Can anyone extrapolate that when one is not prime then both are not prime?
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Old 2003-04-13, 20:53   #5
jocelynl
 
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After more evaluation,

if one is not prime then the other one can be prime. So far I found no mersenne prime in that chaine.
Can there be a mersenne prime M with
N={1,2,3.....}
P=(2^N*5+2^N-1)
M=2^P-1


Joss
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Old 2003-04-13, 23:49   #6
S80780
 
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For another question I just found an answer on Chris Caldwell's PrimeGlossary: do all palindromic primes have a p^n number of digits? - No!
For base-10, I tested it up to 2^31 - 1, which gave 1, 2, 3, 5 and 7 digits.
For twin palindromic primes (BTW, I think 'palindromic prime pair' could be a better name to avoid confusion with palindromic primes that are also prime twins, besides it's a nice anaphora.:)) this isn't the case, either. Counter example: 1009 and 9001.

Benjamin

P.S.: If I got it right, in Chris Caldwell's PrimeGlossary, they are called 'invertable primes'.
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