- This post is about the Wieferich primes, 1093 and

3511.

(1) Concerning the two in concert producing a prime P

(2) Concerning what I have found to be an interesting

composite for P+1 and possibilities for primes close

to it

(3) Concerning 3511 itself, a Wieferich prime.

(1) Let f(x,y) = x^2 - y^2 -2xy.

f(3511,1093) = 7294949 and is prime P.

(2) 7294950 = 2*3*3*5*5*13*29*43 which made it easy to

factorise into its 8 factors, of which 5 are less than

(P+1)^(1/8) and all less than (P+1)^(1/4). This fact

seem to me to make 7294950 fairly interesting as only

6435 previous composites have the property of having 8

prime factors, one in more than a thousand.

(3) f(79,26) = 3511 itself remarkable as 79 = 3*26+1

which makes it a "near" Lucas series with its

characteristic second term 79 = 1mod3 and 1mod(the

first term viz. 26 in this case).

(I have postulated that any prime 1mod10 or -1mod10 is

always expressible with x > 3y in f(x,y), and that x

is the second term of a Fibonacci series T(1) = y,

T(2) = x.)

I have more interesting material but unless this

creates interest, I will spare you it.

John McNamara

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http://im.yahoo.com - John McNamara wrote:
> This post is about the Wieferich primes, 1093 and 3511.

The problem is that given two numbers

one can find any number of features for them.

The important thing is, does any of your message help

the search for the next Wieferich prime, above 46 trillion:

http://www.loria.fr/~zimmerma/records/Wieferich.status - On Tue, 04 September 2001, d.broadhurst@... wrote:
> John McNamara wrote:

There's a distributed hunt for them ongoing.

> > This post is about the Wieferich primes, 1093 and 3511.

> The problem is that given two numbers

> one can find any number of features for them.

> The important thing is, does any of your message help

> the search for the next Wieferich prime, above 46 trillion:

> http://www.loria.fr/~zimmerma/records/Wieferich.status

http://catalan.ensor.org/

The client source is available, so anyone with a compiler can join. My alpha got to ~#5 on their tables a short while back.

Phil

Mathematics should not have to involve martyrdom;

Support Eric Weisstein, see http://mathworld.wolfram.com

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http://www.shopping.altavista.com - --- In primenumbers@y..., John McNamara <mistermac39@y...> wrote:
> (1) Let f(x,y) = x^2 - y^2 -2xy.

Notice that (1) should be read f(x,y) = x^2 - y^2 -xy

> f(3511,1093) = 7294949 and is prime P.

>

> (2) 7294950 = 2*3*3*5*5*13*29*43 which made it easy to

> factorise into its 8 factors, of which 5 are less than

> (P+1)^(1/8) and all less than (P+1)^(1/4). This fact

> seem to me to make 7294950 fairly interesting as only

> 6435 previous composites have the property of having 8

> prime factors, one in more than a thousand.

to get f(3511,1093) = 7294949.

Assuming this, other prime pairs (x,y) exist that

satisfy f(x,y) = P (prime) and P+1 have the property

of having 8 prime factors.

As an example, the prime pair (47,2) gives

P=f(47,2)=2111 prime and P+1=2*2*2*2*2*2*3*11.

Regards

Flavio Torasso - I wrote:
> the search for the next Wieferich prime, above 46 trillion

That's recently gone up to 150 trillion, and climbing fast:

http://www.spacefire.com/numbertheory/wieferich.htm