A171381 Numbers k such that (3^k + 1)/2 is prime.

1, 2, 4, 16, 32, 64

Offset: 1

Author

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

Keywords

• nonn
• hard

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 strong-probable-primes to base 3, so don't test with that base. - Don Reble, Jun 15 2010

From Paul Bourdelais, Oct 13 2010: (Start)

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

Examples

			(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 5-composite 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
		

Links

• Anders Björn and Hans Riesel, Factors of generalized Fermat numbers, Math. Comp. 67 (1998), no. 221, pp. 441-446.
• I. J. Calvo, A note on factors of generalized Fermats numbers, Applied Math. Letters 13, (2000), pp. 1-5. [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)]
• M. F. Hasler and G. Guninski, Eliminating some further terms
• W. Keller, Factors of generalized Fermat numbers found after Bjorn & Riesel
• W. Keller, Prime factors of generalized Fermat numbers F'_m(3) = F_m(3) / 2 and complete factoring status

Crossrefs

Cf. A019434, A093625 (the primes), A138083 (exponents of powers of 2), A028491.

Programs

Magma

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

PARI

is_A171381(n)=ispseudoprime(3^n\2+1)  \\ M. F. Hasler, Oct 02 2012

Extensions

Edited by N. J. A. Sloane, Dec 09 2009

Incorrect terms a(7)-a(15) deleted by Jon E. Schoenfield, Jun 12 2010

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

Edited by N. J. A. Sloane, Jun 12 2010 - Jun 16 2010

Edited by M. F. Hasler, Oct 02 2012