A287614 Primes of the form (1 + x)^y + (-x)^y for some positive x, y.

5, 7, 13, 17, 19, 31, 37, 41, 61, 97, 113, 127, 181, 211, 257, 271, 313, 331, 337, 397, 421, 547, 613, 631, 761, 881, 919, 1013, 1201, 1301, 1657, 1741, 1801, 1861, 1951, 2113, 2269, 2381, 2437, 2521, 2791, 3121, 3169, 3571, 3613, 3697, 4219, 4447, 4513, 4651, 5101, 5167, 5419, 6211

Offset: 1

Author

Juri-Stepan Gerasimov, May 27 2017

Keywords

• nonn

Comments

Conjecture: If x is a positive number and (1 + x)^y + (-x)^y is an odd prime number, then y is other odd prime number or even power of two.

Smallest Mersenne prime (A000668) has n ways to write as (1 + k)^m - k^m for positive k: 3, 7, 127, ...

Examples

			5 (x = 1, y = 2), 7 (1, 3), 13 (2, 2), 17 (1, 4), 19 (2, 3), 31 (1, 5), 37 (3, 3), 41 (4, 2), 61 (3, 4 or 2, 5), 97 (2, 4), 113 (7, 2), 127 (1, 7 or 3, 6), 181 (9, 2), 211 (2, 5), 257 (1, 8), 271 (9, 3).
		

Crossrefs

Cf. A000668, A285929, A283653, A286348.

Programs

Mathematica

mx = 10^4; f[x_, y_] := (1+x)^y + (-x)^y; x=0; Union@ Reap[ While[ f[++x, 2] < mx, y=1; While[(v = f[x, ++y]) < mx, If[PrimeQ@ v, Sow@v]]]][[2, 1]] (* Giovanni Resta, May 31 2017 *)