A072180 Numbers k such that 2^k - k^2 is prime.

5, 7, 9, 17, 19, 51, 53, 81, 83, 119, 189, 219, 227, 301, 455, 461, 623, 2037, 2221, 2455, 3547, 5515, 6825, 8303, 9029, 12103, 49989, 55525, 64773, 80307, 119087, 141915, 192023, 205933, 301683, 307407

Offset: 1

Author

Daniel Gronau (Daniel.Gronau(AT)gmx.de), Jun 30 2002

Keywords

• hard
• nonn

Comments

The numbers corresponding to k = 2037, 2221, 3547 and 5515 have been certified prime with Primo. - Rick L. Shepherd, Nov 10 2002

The remaining k's > 1000 correspond only to probable primes.

Certainly k must be odd. Let N(k) = 2^k - k^2. Additional restrictions come from the facts that 7 | N(k) if k is in {2, 4, 5, 6, 10, 15} mod 21 and 17 | N(k) if k is in {31, 57, 61, 71, 107, 109, 113, 131} mod 136. - Daniel Gronau, Jul 06 2002

Henri Lifchitz found the terms > 40000 in 2001 and 119087 in March 2002. - Hugo Pfoertner, Nov 16 2004

Links

• Henri Lifchitz, Renaud Lifchitz, PRP Top Records 2^n-n^2.

Crossrefs

Cf. A024012, A064539, A075896, A072164.

Programs

Mathematica

Do[ If[ PrimeQ[ 2^n - n^2], Print[n]], {n, 1, 22850, 2}]

PARI

is(n)=isprime(2^n-n^2) \\ Charles R Greathouse IV, Feb 17 2017

Extensions

Edited and extended by Robert G. Wilson v, Jul 01 2002

More terms from Hugo Pfoertner, Nov 16 2004

More terms from Henri Lifchitz submitted by Ray Chandler, Mar 02 2007