A164512 Prime power pairs of form (p^a, q^b = p^a + 1), a >= 1, b >= 1.

2, 3, 3, 4, 4, 5, 7, 8, 8, 9, 16, 17, 31, 32, 127, 128, 256, 257, 8191, 8192, 65536, 65537, 131071, 131072, 524287, 524288, 2147483647, 2147483648, 2305843009213693951, 2305843009213693952, 618970019642690137449562111, 618970019642690137449562112

Offset

1,1

Comments

Consecutive prime powers with positive exponents.

a(n) = Ordered union of {2^3, 3^2, Fermat primes, Fermat primes - 1, Mersenne primes, Mersenne primes + 1}.

It is not known whether this sequence is infinite (but it is believed to be).

2^3, 3^2 are the only consecutive prime powers with exponents >= 2 (this is a consequence of the Catalan-Mihailescu theorem).

Only the first 5 Fermat numbers f_0 to f_4 are known to be prime.

It is conjectured that there exist an infinite number of Mersenne primes.

Links

Daniel Forgues, Table of n, a(n) for n = 1..48

Eric Weisstein's World of Mathematics, Catalan's Conjecture.

Eric Weisstein's World of Mathematics, Mersenne Prime.

Eric Weisstein's World of Mathematics, Fermat Prime.

Crossrefs

Cf. A019434 (Fermat primes), A000668 (Mersenne primes).

Sequence in context: A070046 A130120 A204892 * A127434 A266475 A205402

Adjacent sequences: A164509 A164510 A164511 * A164513 A164514 A164515

Keyword

hard,nonn

Author

Daniel Forgues, Aug 14 2009

Extensions

Edited by N. J. A. Sloane, Aug 24 2009

Status

approved