--- In

primenumbers@yahoogroups.com, "julienbenney" <jpbenney@...> wrote:

>

> It's odd I just came to think of primes as I came in after the first

rain in Melbourne for weeks.

Glad to see your part of Australia is finally getting some rain. It's

pretty dry down there I hear.

I just noticed something in common with all the prime and probable

prime repunits: R(2), R(19), R(23), R(317), R(1031), R(49081),

R(86453) and R(109297).

They are all prime of course, but all of their values mod (10^x) are

also a prime, a power of 3 times a prime, or one.

For instance take the largest prp prime repunit R(109297):

109297 is prime

109297 mod 100000 = 9297 = 3^2 * 1033

109297 mod 10000 = 9297 = 3^2 * 1033

109297 mod 1000 = 297 = 3^3 * 11

109297 mod 100 = 97

109297 mod 10 = 7

Furthermore, notice that with the derived 1033 prime factor above, it

too has all its values mod 10^x as prime, a power of 3 times a prime,

or one.

And so on with the rest of the prime repunits. Coincidence? I doubt

it would hold for all larger prime repunits, but it's fun observing

what is likely the law of small numbers in effect.

Mark