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

[cino hilliard]

>From: zak seidov <zakseidov at yahoo.com>
>To: seqfan at ext.jussieu.fr
>Subject: Primes of the form (3^n + 5^n)/2.
>Date: Sat, 9 Sep 2006 23:43:00 -0700 (PDT)
>
>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?


Hi Zak and others,

The Law of Diminishing Primes is the foe here. Consider

As we all know, the probability that a random number n is prime ~ 1/log(n).  
So this
expression -> 0 as n ->inf.

Law of Diminishing Primes
------------------------------------------------
The yield of prime numbers by additional increments to the parameters of a 
prime generating
formula which experiences expontial output will reach a point where larger 
and larger increments
will produce fewer and fewer primes.:-)

Therefore,

The search for more and more primes numbers by incremental input into a 
formula that grows
exponentially is futile.

I submitted the listing below which this post inspired. I invite Zak as 
co-author. Only 2 of the trials
up to a=43,b=47 yield 3 primes for n <= 1000:
a=3,b=5
a,b,n,S(n)
3,5,2,17
3,5,4,353
3,5,8,198593

and
a=7,b=11.
7,11,4,8521
7,11,16,22991481397070881
7,11,32,1055689389481464764733130779754561

So it is doubtful the formula (a^n+b^n)/2 will ever meet the OEIS criterion 
for inclusion for any
consecutive primes a and b.

Similarly, if a and b are consecutive odd numbers the sequence breaks down 
at a=33,b=35
n<=1000. Here inclusion to OEIS is possible for
a=1,b=3,n=1 whicih is A093625

Subject: NEW SEQUENCE FROM Cino Hilliard
%I A000001
%S A000001 
17,37,8521,21601,229,106921,205081,289278699121,815401,1398841,3122281,2029
%N A000001 First occurrence of (p1^n + p2^n)/2 is prime where p1 and p2 are 
consecutive primes.
%C A000001 These numbers are all of the form 4n+1.
The next term for a=37,b=41,n <=1000,is
8274567108488469403564696641244659777685186165444353190460129729940809291805549571887038803603334751361,
The next term a=41,b=43 is 3122281 n<=1000
The next term a=43,b=47 is 2029    n<=1000
The next term for a=47,b=53 is not found for n <=1000
The Law of Diminishing Primes finally invokes itself here.
%D A000001 Zak Seidov query about (3^n + 5^n)/2 is prime in seqfan 9/10/06
%o A000001 (PARI) g(n,a,b) = 
for(x=1,n,y=(a^x+b^x)/2;if(ispseudoprime(y),print(a","b","x","y)))
%O A000001 1,1
%K A000001 ,nonn,
%A A000001 Cino Hilliard (hillcino368 at hotmail.com), Sep 10 2006

Have Fun,
Cino