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

Author

Daniel Forgues, Aug 14 2009

Keywords

• hard
• nonn

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).

Extensions

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