bill@ur-cvsvax.UUCP (Bill Vaughn) (11/18/84)
Since the subject of palindromic primes has come up, I thought I would
share some of the results I came up with about 5 years ago with a
Fortran program on an 11/34. I computed all the palindromic primes
between 2 and 999,999,999. I division routine itself was written in
MACRO-11.
Anyway here is a table of the results. (PP=palindromic prime;
Pn=palindromic number;
#digits #numbers #primes #Pn's #PP's
3 900 143 (15.9%) 90 15 (16.7%)
5 90000 8363 (9.3%) 900 93 (10.3%)
7 9000000 586080 (6.5%) 9000 668 (7.4%)
9 900000000 45086079 (5.0%) 90000 5172 (5.7%)
It seems that the ratio of PP's wrt all Pn's follows very closely with
the ratio of primes wrt all numbers. Does this mean that one should be
able to prove a Palindromic Prime Number Theorem? It's one of those
conjectures that's painfully obvious, but where's the beef. :-) (proof)
In a previous article, I mentioned that I looked for specific kinds of PP's
i.e. those with only 2 different digits in their representation. I forgot
to mention that my search was even more restrictive than that. I looked
for 'alternating' PP's and 'constant middle' PP's, that is, PP's of the
form 'aba...aba' or 'abb...bba' where a and b are relatively prime digits.
It turns out that there are only 30 possibilities of each kind within any
decade in which there are any Pn's. The search really starts with 5 digit
PP's since 3 digit PP's trivially satisfy both criteria.
Here are the gems I found among the dust:
18181 32323 35353 72727 74747 78787 94949 95959
13331 15551 16661 19991 72227 75557 76667 78887 79997
1212121 1616161
1333331 1444441 1777771 3222223 3444443 7666667 9222229 9888889
323232323 383838383 727272727 919191919 929292929 979797979 989898989
188888881 199999991 322222223 355555553 722222227
All these numbers are prime. Sigh. Chaos and order inextricably interwoven.
Bill Vaughn
Univ. of Rochester (CVS)
Rochester, NY 14627
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