- It's odd I just came to think of primes as I came in after the first rain in Melbourne for weeks.

I was reminded of the proof of the primality of R317 and just looked up "repunit prime" on

Wikipedia, expecting to find nothing at all that I did not already know.

When I checked it, however, I found, to my greatest surprise, that Harvey Dubner had found

yet another probable repunit prime with 109297 copies of the number one. It was Dubner

who found, after a lengthy search when for a long time he did not expect to find a repunit

prime, the previous probable prime repunits R49081 and R86453.

The distance between R1031 and R49081 and the fact that there seemed to be three "pairs"

of repunit primes or PRPs (19 and 23, 317 and 1031, 49081 and 86453) made me think

intuitively that the next PRP repunit would be extremely large. I imagined when I first saw

R86453 to be a probable prime that the next PRP repunit would have over one million digits -

sometimes imagining it to be one of the numbers of the prime quadruples beginning with

1,006,301 and 1,006,331. Having seen R109297, I can imagine the next PRP repunit having

several million digits - Dubner himself intends only going to 200,000 and I do not know what

to say if there is another PRP repunit up to that small a number of digits. - From: julienbenney

Sent: 22 April 2007 14:32

>Maybe they should search other bases. I recall reading tables of

Phi(4229,51) is PrP. The first base for which I have not yet found a PrP

>generalised repunit primes for bases up to 51 (for base 51, there were

>none at all up to the limit of n=2609).

is 152. There are 34 others under 1000 (excepting perfect powers)

without a PrP, all have been searched for (prime) exponents up to 2^15.

I can find no hints as to covering sets that would exclude any of these.

>For instance, what is the next number in the sequence 3, 43, 73, 487,

37441 is a candidate. The answer lies in (10177,37441].

>2579, 8741, ...??

(Base-15 PrP GRUs.)

I should do a OEIS search and update whichever sequences I can. Last

time I checked, I had "first base for each exponent" and "first exponent

for each base" considerably improved from the published sequences, but

that was years ago and someone else may have been working on them.

There's too much "real life" going on atm ;-)

Best Regards,

Andy Steward