A111635 Smallest prime of the form x^(2^n) + y^(2^n) where x,y are distinct integers.

2, 5, 17, 257, 65537, 3512911982806776822251393039617, 4457915690803004131256192897205630962697827851093882159977969339137, 1638935311392320153195136107636665419978585455388636669548298482694235538906271958706896595665141002450684974003603106305516970574177405212679151205373697500164072550932748470956551681

Offset: 0

Author

Max Alekseyev, Aug 09 2005

Keywords

• nonn

Comments

Is this sequence defined for all n?

From Jeppe Stig Nielsen, Sep 16 2015: (Start)

Numbers of this form are sometimes called extended generalized Fermat numbers.

If we restrict ourselves to the case y=1, we get instead the sequence A123599, therefore a(n) <= A123599(n) for all n. Can this be an equality for some n > 4?

The formula x^(2^m) + y^(2^m) also gives the decreasing chain {A000040, A002313, A002645, A006686, A100266, A100267, ...} of subsets of the prime numbers if we drop the requirement that x != y and take all primes (not just the smallest one) with m greater than some lower bound.

(End)

For more terms (the values of max(x,y)), see A291944. - Jeppe Stig Nielsen, Dec 28 2019

Links

• Jeppe Stig Nielsen, Table of n, a(n) for n = 0..9

Crossrefs

Cf. A019434, A100270, A123599, A291944.