# (pp-1)/2 is prime: {27, 2187,...}

Robert G. Wilson v rgwv at kspaint.com
Tue Oct 15 01:10:05 CEST 2002

```Dean,

Here they are to 10^7.

Sincerely,
Bob.

Dean Hickerson wrote:

>Zakir F. Seidov (seidovzf at yahoo.com) wrote:
>
>>with my misery "database" of 1111 perfect primes < 1,000,000
>>i've found only two pp: {27, 2187} such that (pp-1)/2 is prime.
>>
>>can anybody provide me next 1000 pp's and/or find several next pp's in
>>subject. thanks, zak
>>
>
>I asked him what he meant by "perfect primes" and he explained that it was
>a typo for "perfect powers", i.e. numbers a^b with integers a>=1 and b>=2.
>
>So suppose that  (a^b-1)/2  is prime.  Since  a-1  divides  a^b-1,  we must
>have  a=3.  Also, if  b  is composite, say  b=c*d  with  c>1  and d>1,  then
>(3^c-1)/2  divides  (3^b-1)/2.  Hence  b  must be prime.
>
>The values of  b  for which  (3^b-1)/2  is prime are given in A028491;
>the first several are:
>
>    3, 7, 13, 71, 103, 541, 1091, 1367, 1627, 4177, 9011, 9551
>
>The corresponding primes  (3^b-1)/2  are:
>
>    13, 1093, 797161, 3754733257489862401973357979128773, ...
>
>These weren't in the OEIS, so I've submitted them.
>
>Dean Hickerson
>dean at math.ucdavis.edu
>

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