

A164512


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


2



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
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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 if 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 CatalanMihailescu 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
Weisstein, Eric W., Catalan's Conjecture
Weisstein, Eric W., Mersenne Prime
Weisstein, Eric W., Fermat Prime


CROSSREFS

Cf. A019434 Fermat primes: primes of form 2^(2^n) + 1, n >= 0.
Cf. A000668 Mersenne primes (of form 2^p  1 where p is a prime).
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



