unbent@ecsvax.UUCP (11/13/84)
==> My colleague, Paul Ziff, is a collector of both patterns and prime numbers. Recently these two avocations intersected in his noticing *palindromic primes*, i.e., primes whose decimal representations read the same forward and backward: 11, 101, 151, 757, for example. He fired up his little house computer and set off in search of them. Evidently he came up with quite a few, before running out of computational ability. In the process, he noticed something else: There weren't any 4-digit, 6-digit, or 8-digit palindromic primes. Maybe he got beyond that, to 9- and/or 10-digit primes, because he got far enough, in any event, to arrive at a conjecture: Ziff's conjecture: The number of digits in the decimal representation of a palindromic prime is itself prime. He asked me if I had any idea (1) whether anything like that conjecture was known to be true, and (2) how in the world one might go about trying to establish a strange conjecture like that. [It's strange because it attempts to interrelate facts about *numbers* with facts about (decimal) *numerals*.] I didn't know the answer to either (1) or (2), but I told him I'd put it on the net -- and here it is. Anyone out there have any ideas? --Jay Rosenberg Dept. of Philosophy ...mcnc!ecsvax!unbent Univ. of North Carolina Chapel Hill, NC 27514
FtG@rochester.UUCP (Gary L. Peterson) (11/14/84)
ecsvax!unbent asks if prime palindromes must have prime length.
The following shows that the length cannot be even:
Fact 1: all numbers of the form 10**odd + 1 are divisible by 11.
Fact 2: all even length palindromes are of the form
a1 (10**k + 1) + a2 * (10**(k-2) + 1) * 10 + ... a(k/2+1) *11 * 10**k/2
Where k = length -1 (= odd) and ai's are digits 0..9 with a1<>0.
Since each term is divisible by 11, the whole number is.
Generalization: Any even length palindrome number in base b is divisible
by b+1.
Next week: odd numbers!
FtG@rochester
> ==>chuck@dartvax.UUCP (Chuck Simmons) (11/15/84)
> There weren't any 4-digit, 6-digit, or > 8-digit palindromic primes. Maybe he got beyond that, to 9- and/or 10-digit > primes, because he got far enough, in any event, to arrive at a conjecture: > > Ziff's conjecture: The number of digits in the decimal representation of a > palindromic prime is itself prime. Note that any palindromic number with an even number of digits is divisible by 11, whether prime or not. Ziff obviously didn't get to 9-digit primes. I just asked our honeywell to find me a 9-digit palindromic prime. It liked the 4th one it looked at: 100030001 dartvax!chuck
lambert@mcvax.UUCP (Lambert Meertens) (11/16/84)
100030001 is a palindromic prime number of composite length.
--
Lambert Meertens
...!{seismo,philabs,decvax}!lambert@mcvax.UUCP
CWI (Centre for Mathematics and Computer Science), Amsterdambill@ur-cvsvax.UUCP (Bill Vaughn) (11/16/84)
11 is the only palindromic prime with an even number of digits. All other
even palindromic numbers are divisible by 11. This is because the
alternating sum of the digits of a palindromic number with an even number
of digits is 0, and this is a well knwon test for divisibilty by 11.
As for Dr. Ziff's conjecture, I think that it is highly unlikely. I
probably have a counter-example of some 9 digit palindromic prime somewhere
in some research I did some years ago, but I can't put my hands on it
right at this moment. I did find these facts however: the number of n-digit
palindromic numbers is 9*10^[(n-1)/2]. (PP = palindromic primes).
There are 15 3 digit PP's out of 90 possibilities. ~17%
" " 93 5 digit PP's " " 900 " ~10%
" " 668 7 digit PP's " " 9000 " ~7%
Even though I don't have the number of 9 digit PP's out of the 90,000
possibilities, it just doesn't seem possible that they could buck this
trend and go to zero percent.
Some of the invetigations I carried out years ago showed that even if
one restricted oneself to palindromic's which use only 2 different digits
(i.e 18181 35353 7474747474 etc.) there are still plenty of primes out
there. I only used Fermat's test with several different bases however,
so the best I can say is that there are plenty of pseudo-primes for as
far as you care to look. I think I went up to about 19-21 digits.
These ramblings lead me to ask a question which has been nagging me for
some time:
Are there an infinite number of palindromic primes? If so, can someone
point me to a proof or sketch one out.
Would this proof(if it exists) apply to subsets of PP's such as those which
use only 2 different digits?
What about other bases?
Bill Vaughn
Univ. of Rochester
Center for Visual Science
{allegra,seismo,decvax}!rochester!ur-cvsvax!bill