> Paul Leyland askedme the other day about 4^n-3. As far as I know the

> biggest "ordinary prime" (whatever that is) is 4^7057-3 proven by

> Preda Milhailescu.

Indeed, as I am toying with the idea of searching for further examples

and putting up a web page to coordinate the search, should anyone else

be interested in joining in. However, pressure of other work means that

it won't happen in the immediate future 8-(

> I am intersted in these because I have never found a composite of

> the form F:2^N-2^k-1 for which F|2^F-2. In the above case k=1 and

> n=2N.

>

> Modular reducton of a 2N bit number is particularly easy since

> 2^N=2+1 and hence requires just two shifts and additions. Moreover to

> test F we could check 2^(2^N)=8 modulo F i.e take N (FFT) squarings

> of 2, use the simple modular reduction, and finally check to see if

> this equals 8.

>

> I checked an old email file to the old utm list server for back on

> the 3rd May 1999.I wrote:

>

> >4^n-3 is proven prime. n:

[Lists deleted.]

Thanks Paul! This will help populate the web page, and also gives a few

nice test cases to pick up really stupid bugs.

Paul