## 18883Re: R109297

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• Apr 23, 2007
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--- 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
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