# two forms of "Cunningham's prime pairs"

Wouter Meeussen eu000949 at pophost.eunet.be
Tue Aug 24 00:04:08 CEST 1999

```hi all,

if we define as "Cunningham Prime pairs" the primes "b" and "e"
(for *b*ase and *e*xponent) such that (b^e -1)/(b-1) is prime, and
either  e is minimal for the (given) b
or      b is minimal for the (given) e

then we get: {b,e}=

either
{{2,2},{3,3},{5,3},{7,5},{11,17},{13,5},{17,3},{19,19},{23,5},{29,5},{31,7},
{37,13},{41,3},{43,5},{47,127},{53,11},{59,3},{61,7},{67,19},{71,3},{73,5},
{79,5},{83,5},{89,3},{97,17},{101,3},{103,19},{107,17},{109,17},{113,23},
{127,5},{131,3}}

or
{{2,2},{2,3},{2,5},{2,7},{5,11},{2,13},{2,17},{2,19},{113,23},{151,29},{2,31},
{61,37},{53,41},{89,43},{5,47},{307,53},{19,59},{2,61},{491,67},{3,71},
{11,73},{271,79},{41,83},{2,89},{271,97},{359,101},{3,103},{2,107},
{79,109},{233,113},{2,127},{7,131}}

reverting to the PrimePi (-index) of the relevant primes gives resp. :

{1,2,2,3,7,3,2,8,3,3,4,6,2,3,31,5,2,4,8,2,3,3,3,2,7,2,8,7,7,9,3,2}
and
{1,1,1,1,3,1,1,1,30,36,1,18,16,24,3,63,8,1,94,2,5,58,13,1,58,72,2,1,22,51,1,4}

Remark:
The Cunningham group-effort (Like Woltman's Mersenne search)
goes for very large exponents, but limits itself to bases up to 12.
I'm cooking up some large "minimal exponents" for larger bases right now.

wouter.

(* Mma expressions used *)
Table[Flatten@{b=Prime[n],Select[Prime[Range[32]],
PrimeQ[(b^#-1)/(b-1)]&,1]},{n,32}]

Table[Flatten@{e=Prime[n];Select[Prime[Range[100]],
PrimeQ[(#^e-1)/(#-1)]&,1], e},{n,32}]

```