- --- In primenumbers@yahoogroups.com, "djbroadhurst" <d.broadhurst@...> wrote:
>

Congrats on establishing that substantial sequence, David.

> Prime q=2*p+1 with primes b<c<p such that q^2|b^p-1 and q^2|c^p-1

>

> 555383, 1767407, 2103107, 7400567, 12836987, 14668163, 15404867,

> 16238303, 19572647, 25003799, 26978663, 27370727, 35182919,

> 36180527, 38553023, 39714083, 52503587, 53061143, 53735699,

> 55072427, 63302159, 70728839, 77199743, 77401679, 86334299,

> 97298759, 97375319, 103830599, 106208783, 106710287, 108711599,

> 112590683, 120441239, 124581719, 126236879, 128538659, 129881603,

> 133833983, 141132143, 141194387, 145553399, 151565087

>

> Comments: (p,q) is a Sophie Germain prime pair; (b,q) and (c,q)

> are Wieferich prime pairs; each of (b,c) is a square modulo q^2.

> The sequence is now complete up to the 42nd term, q=151565087.

> Mike Oakes set a puzzle on a more general case with primes such

> that b<p1<q, c<p2<q, b<c, q^2|b^p1-1 and q^2|c^p2-1. His sequence

> is complete only up q=27370727, containing only two new known

> primes, q=2452757 and q=22796069, with p1=p2=(q-1)/4, found in

> http://tech.groups.yahoo.com/group/primenumbers/message/23175 by

> a simple analysis of http://www.cecm.sfu.ca/~mjm/WieferichBarker

It must be worthy of submission to Neil's OEIS.

Your upper limit on q is about 5 times higher that I have reached for the "more general case" - which my code would take about 5^2*4=100 days on one 3.6GHz core to complete.

Mike - Hi , all ,

Not even draft yet , let alone published :

http://upforthecount.com/math/nnp.html

still needs a lot of work .

But , I have updated the top of the main table :

http://upforthecount.com/math/nnp2.txt

with complete factorizations now extending through

np ( 90 ) .

When is np ( n ) prime ?

So far , only for 1 , 2 and 3 .

When n mod 6 = 4 , np ( n ) has at least 3 algebraic

factors , 3 and n^2 + n + 1 ( twice ) .

But what about the residue ( primitive part ? ) ?

When is this prime ?

n t n^n + (n+1)^(n+1)

4 4 3 * 7 * 7 * 23

16 6 3 * 7 * 7 * 13 * 13 * 34041259347101651

Any others ?

Cheers ,

Walter