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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 (list; graph; refs; listen; history; text; internal format)
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 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

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

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Last modified October 19 07:21 EDT 2018. Contains 316337 sequences. (Running on oeis4.)