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

[Dean Hickerson]

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