

A171381


Numbers k such that (3^k + 1)/2 is prime.


7




OFFSET

1,2


COMMENTS

Note that k must be a power of 2 (cf. A138083).
Similar to Fermat primes (A019434), and for the same reasons we expect this sequence to be finite as well.
The numbers (3^k + 1)/2 are strongprobableprimes to base 3, so don't test with that base.  Don Reble, Jun 15 2010
Terms in sequence (3^k + 1)/2 factored to 10^18:
(3^(2^21)+1)/2 has factors: 155189249
(3^(2^22)+1)/2 is composite: RES64: [A158D7ED3E1CC427] (425462 sec)
(3^(2^23)+1)/2 is composite: RES64: [B0F07A3D55C5082A] (3424080 sec)
(3^(2^26)+1)/2 has factors: 3221225473
(3^(2^28)+1)/2 has factors: 12348030977
(3^(2^29)+1)/2 has factors: 77309411329
(3^(2^31)+1)/2 has factors: 4638564679681
(3^(2^32)+1)/2 has factors: 206158430209
(3^(2^34)+1)/2 has factors: 50474455662593
(3^(2^36)+1)/2 has factors: 911220261519361
(3^(2^38)+1)/2 has factors: 6597069766657
(3^(2^39)+1)/2 has factors: 46179488366593
(3^(2^44)+1)/2 has factors: 15586676835352577
(3^(2^45)+1)/2 has factors: 16044073672507393
(3^(2^49)+1)/2 has factors: 7881299347898369
(3^(2^51)+1)/2 has factors: 891712726219358209
(3^(2^54)+1)/2 has factors: 180143985094819841
For all other k < 55 (specifically, k = 24, 25, 27, 30, 33, 35, 37, 40, 41, 42, 43, 46, 47, 48, 50, 52, 53), no factor < 10^18 has been found.
(End)
Also, numbers k such that 3^k+1 is a semiprime.  Sean A. Irvine, Oct 16 2023


LINKS

I. J. Calvo, A note on factors of generalized Fermats numbers, Applied Math. Letters 13, (2000), pp. 15. [Gives divisibility criteria for 3^(2^m)+1 by primes of the form p=3*2^n+1 (p=7, 13, 97, 193 ...) (Theorem 2.1) and for primes of this form when they are divisors of Fermat numbers (Theorem 2.2)]


EXAMPLE

(3^(2^0)+1)/2 = (3^1+1)/2 = 2 is prime.
(3^(2^1)+1)/2 = (3^2+1)/2 = 5 is prime.
(3^(2^2)+1)/2 = (3^4+1)/2 = 41 is prime.
(3^(2^3)+1)/2 = (3^8+1)/2 = 3281 is divisible by 17=1+2^4.
(3^(2^4)+1)/2 = (3^16+1)/2 = 21523361 is prime.
(3^(2^5)+1)/2 = (3^32+1)/2 = 926510094425921 is prime.
(3^(2^6)+1)/2 = (3^64+1)/2 = 1716841910146256242328924544641 is prime.
(3^(2^7)+1)/2 = (3^128+1)/2 is divisible by 257=1+2^8, so 2^7 is not a term.
(3^(2^8)+1)/2 = (3^256+1)/2 is divisible by 1+2^9*24, so 2^8 is not a term.
(3^(2^15)+1)/2 is divisible by 2^(2^4)+1, so 2^15 is not a term.  Georgi Guninski, Jun 13 2010
(3^(2^19)+1)/2 is divisible by 13631489, so 2^19 is not a term.  Paul Zimmermann, Jun 14 2010
(3^(2^20)+1)/2 is 5composite so 2^20 is not a term.  Serge Batalov, Jun 14 2010
According to PFGW, 2^20 is not in the sequence: PFGW Version 3.3.4.20100405.x86_Stable [GWNUM 25.14] (3^1048576+1)/2 is composite: RES64: [9EE4CA1AABB9A816] (3229 sec) Base 5. Base 3 is useless here [cf. comment by Don Reble  Ed.].  Georgi Guninski, Jun 15 2010
2^23 is not in the sequence (listed as "Composite but no factor known" on the second Keller link).  Serge Batalov, Jun 18 2010
Verified 2^23 yields a composite: base 2 (PFGWv3.3.1).  Paul Bourdelais, Apr 25 2011


PROG

(Magma) IsPrime((3^(2^15)+1) div 2); // shows that 15 is not a term  Jon E. Schoenfield, Jun 13 2010


CROSSREFS



KEYWORD

nonn,hard


AUTHOR

Joao Carlos Leandro da Silva (zxawyh66(AT)yahoo.com), Dec 07 2009


EXTENSIONS

The next term, if it exists, is at least 2^19.  Georgi Guninski, Jun 13 2010
A comment in A093625 from Don Reble, Apr 28 2004, says the next term, if it exists, is >= 2^21.
k=2^21 yields a number divisible by 1+2^22*37.  M. F. Hasler, Jun 14 2010


STATUS

approved



