## Re: Square factors of b^p-1

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• ... In the course of this Mike Oakes found 8 pairs of Wieferich pairs not recorded Michael Mosinghoff, who condidered only q = 1 mod 4 when q 10^7 and hence
Message 1 of 81 , Oct 3, 2011
"mikeoakes2" <mikeoakes2@...> wrote:

> > Puzzle 3a: Find primes b1 < p1 < q, b2 < p2 < q, b1 < b2, s.t.
> > b1^p1 = b2^p2 = 1 mod q^2, (with no limits on q^2).
>
> For b1, b2, q < 10^7.5, the complete list of such q is:-
> 555383
> 1767407
> 2103107
> 2452757
> 7400567
>
> 12836987
> 14668163
> 15404867
> 16238303
> 19572647
> 22796069 [found by DB]
> 25003799
> 26978663
> 27370727
>
> Each (q-1) is of the form 2^m*p, and p1 = p2 in each case.

In the course of this Mike Oakes found 8 pairs of
Wieferich pairs not recorded Michael Mosinghoff,
who condidered only q = 1 mod 4 when q > 10^7 and
hence missed SG pairs (p,q=2*p+1).

In the format [q,m,p,[b1,b2]] these new
double Wieferich results are:

[12836987, 1, 6418493, [2061197, 4631743]]
[14668163, 1, 7334081, [3692407, 7145591]]
[15404867, 1, 7702433, [2824163, 3123217]]
[16238303, 1, 8119151, [639167, 1005581]]
[19572647, 1, 9786323, [3204463, 7873399]]
[25003799, 1, 12501899, [59273, 4701583]]
[26978663, 1, 13489331, [2589553, 8582417]]
[27370727, 1, 13685363, [5002507, 5356201]]

Moreover M.O must have found many more
isolated Wieferich pairs with q > 10^7.
Perhaps these might be communicated to M.M ?

David
• 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
Message 81 of 81 , Nov 29, 2011
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
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