[seqfan] Re: Addition to A125854

Max Alekseyev maxale at gmail.com
Wed Jan 21 15:35:13 CET 2009

On Tue, Jan 20, 2009 at 6:54 PM, Peter Luschny
<peter.luschny at googlemail.com> wrote:
> Max Alekseyev provided A125854(4) = 2001907169
> quite recently and added the comment
> "No other terms below 10^10."
> Max, how did you check this?

Well, first of all, it should be really A125854(5) = 2001907169 (it
was a typo in the index).

I did search using a new criteria that I've added to A125854:

%C A125854 Note that for a prime p, H([p/2]) == 2*(2^(-p(p-1))-1)/p^2
(mod p). Therefore
               a prime p divides the Wolstenholme number
A001008((p+1)/2) if and
               only if 2^(-p(p-1)) == 1 - p^2 (mod p^3).

In PARI/GP, this can be implemented as:

forprime(p=3,10^10,if( (2+O(p^2))^(-p*(p-1))==1-p^2, print(p)) )

I've also extended the search up to 10^11 but no found new terms.
The entry A125854 will be updated soon.


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