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%I #19 Feb 26 2024 09:13:36
%S 5,7,13,17,19,31,37,41,61,97,113,127,181,211,257,271,313,331,337,397,
%T 421,547,613,631,761,881,919,1013,1201,1301,1657,1741,1801,1861,1951,
%U 2113,2269,2381,2437,2521,2791,3121,3169,3571,3613,3697,4219,4447,4513,4651,5101,5167,5419,6211
%N Primes of the form (1 + x)^y + (-x)^y for some positive x, y.
%C 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.
%C Smallest Mersenne prime (A000668) has n ways to write as (1 + k)^m - k^m for positive k: 3, 7, 127, ...
%e 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).
%t 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 *)
%Y Cf. A000668, A285929, A283653, A286348.
%K nonn
%O 1,1
%A _Juri-Stepan Gerasimov_, May 27 2017
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