Primes of the form (3^n + 5^n)/2.

[David Wilson]

So this problem is similar to the Fermat prime problem.

Do we expect only a finite number of Fermat primes?

----- Original Message ----- 
From: "Edwin Clark" <eclark at math.usf.edu>
To: "Hugo Pfoertner" <all at abouthugo.de>
Cc: <seqfan at ext.jussieu.fr>
Sent: Sunday, September 10, 2006 11:12 AM
Subject: Re: Primes of the form (3^n + 5^n)/2.


> On Sun, 10 Sep 2006, Hugo Pfoertner wrote:
> 
>> zak seidov wrote:
>> > 
>> > Not appeared in OEIS:
>> > 
>> > %S A000001 17, 353, 198593
>> > %N A000001 Primes of the form (3^n + 5^n)/2.
>> > %C A000001 Corresponding n's are 2^1, 2^2, and 2^3.
>> > What are next terms?
>> > Cf. A074606 3^n + 5^n.
>> > %A A000001 Zak Seidov, Aug 27 2006
>> > 
>> > My kind request to gurus:
>> > What are next terms?
>> > Thanks, Zak
>> 
>> If any more should exist, they are > (3^2500+5^2500)/2
>> 
>> Hugo
>> 
> 
> Since x^n + y^n has x+y as a factor if n is odd, we can assume
> that n is a power of 2. Maple shows that up to n = 2^15, there
> are no more primes of the form (3^n + 5^n)/2. 
> 
> This raises the question:
> Is it true that x^n + (x+2)^n is irreducible over Q for n a power
> of 2? 
> 
> --Edwin
> 
> 
> 
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