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• In a message dated 05/09/03 20:56:38 GMT Daylight Time, ... Mark: You have left out of consideration the (just as interesting) case when p is not a prime but
Message 1 of 5 , Sep 10, 2003
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In a message dated 05/09/03 20:56:38 GMT Daylight Time,
mark.underwood@... writes:

> Here's a relation that is well known due to Fermat's Last Theorem:
>
> b^p + a^p
>
> where p is prime and a and b are relatively prime.
>
> Divide this relation by (b+a) and you have a potential prime.

Mark: You have left out of consideration the (just as interesting) case when
p is not a prime but is a power of 2.

In that case, you don't/can't divide by (b+a), and F(a,b,m) = b^(2^m) +
a^(2^m) is itself potentially prime.

If a = 1, b = 2, these are the Fermat primes.

If a = 1, b <> a+1, they are "Generalized Fermat" primes.

If a > 1, b = (a+1) we tie in with the case that you singled out in your
posts.
Back in Aug2000 I did an exhaustive search of these forms, in fact, for 1 <=
a <= 1023, 2 <= 2^m <= 8192, and this range had just one (probable) prime
bigger than 10000 digits: 312^4096+311^4096 (10217 digts).
(Unfortunately at that time the lower cutoff for submission to Henri
Lifchitz's database was 12000 digits, so it just languished in a pile of other PrPs,
and it fell to Henri to rediscover it - plus a couple of much bigger ones - for
his database a couple of years later.)

At the same time I searched F(a,b,m) for 1 <= a < b <= 100, 4 <= 2^m <= 8192,
and this range has again just one PrP bigger than 10000 digits:
72^8192+43^8192 (15,216 digits).

Perhaps it's time for someone to extend these investigations with today's
much faster machines?

Mike

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