A121824 Primes of the form (3^n + 5^n)/2.

17, 353, 198593

Offset: 1

Author

Zak Seidov, Aug 27 2006

Keywords

• hard
• nonn
• bref

Comments

Corresponding n are 2^1, 2^2, 2^3. What are the following terms? Cf. A074606 3^n + 5^n.

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? - W. Edwin Clark, Sep 10 2006

Next term, if it exists, is > (3^2500+5^2500)/2. - Hugo Pfoertner, Sep 10 2006

No more terms <= (3^(2^17)+5^(2^17))/2=(3^131072+5^131072)/2. Hence the next term, if it exists, is greater than 10^91616 (so is too large to include). - Herman Jamke (hermanjamke(AT)fastmail.fm), Dec 31 2007

Crossrefs

Cf. A074606, A121710.

Programs

Mathematica

Select[Table[(3^n + 5^n)/2,{n,100}],PrimeQ] (* James C. McMahon, Oct 22 2024 *)

PARI

for(n=1,17, m=(3^(2^n)+5^(2^n))/2;if(isprime(m),print1(m","))) \\ Herman Jamke (hermanjamke(AT)fastmail.fm), Dec 31 2007

Extensions

Edited by N. J. A. Sloane, Jan 13 2008