OFFSET
1,1
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, ...
EXAMPLE
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).
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 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Juri-Stepan Gerasimov, May 27 2017
STATUS
approved