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A286348 Numbers n such that 4^n + (-3)^n is prime.

%I A286348 #19 Sep 08 2022 08:46:19
%S A286348 0,3,4,7,16,17,59,283,311,383,499,521,541,599,1193,1993,2671,7547,
%T A286348 24019,46301,48121,68597,91283,131497,148663,184463,341233
%N A286348 Numbers n such that 4^n + (-3)^n is prime.
%C A286348 Numbers n such that (1 + k)^n + (-k)^n is prime:
%C A286348 0 (k = 0);
%C A286348 A285929 (k = 1);
%C A286348 A283653 (k = 2);
%C A286348 this sequence (k = 3);
%C A286348 0, 2, 3, 4, 43, 59, 191, 223, ... (k = 4);
%C A286348 0, 2, 5, 8, 11, 13, 16, 23, 61, 83, ...(k = 5);
%C A286348 0, 3, 4, 7, 16, 29, 41, 67, ... (k = 6);
%C A286348 0, 2, 7, 11, 16, 17, 29, 31, 79, 43, 131, 139, ... (k = 7);
%C A286348 0, 4, 7, 29, 31, 32, 67, ... (k = 8);
%C A286348 0, 2, 3, 4, 7, 11, 19, 29, ... (k = 9);
%C A286348 0, 3, 5, 19, 32, ... (k = 10);
%C A286348 0, 3, 7, 89, 101, ... (k = 11);
%C A286348 0, 2, 4, 17, 31, 32, 41, 47, 109, 163, ... (k = 12);
%C A286348 0, 3, 4, 11, 83, ... (k = 13);
%C A286348 0, 2, 3, 4, 16, 43, 173, 193, ... (k = 14);
%C A286348 0, 43, ... (k = 15);
%C A286348 0, 4, 5, 7, 79, ... (k = 16);
%C A286348 0, 2, 3, 8, 13, 71, ... (k = 17);
%C A286348 0, 1607, ... (k = 18);
%C A286348 ...
%C A286348 Primes of the form (1 + n)^(2^n) + n: 5, 83, 65539, 7958661109946400884391941, ...
%C A286348 Numbers m such that (1 + k)^m + (-k)^m is not odd prime for k =< m: 0, 1, 15, 18, 53, 59, 106, 114, 124, 132, 133, 143, 177, 214, 232, 234, 240, 256, ...
%C A286348 Conjecture: if (1 + y)^x + (-y)^x is a prime number then x is zero, or an even power of two, or an odd prime number.
%e A286348 3 is in this sequence because 4^3 + (-3)^3 = 37 is prime.
%e A286348 4 is in this sequence because 4^4 + (-3)^4 = 337 is prime.
%t A286348 Select[Range[0, 3000], PrimeQ[4^# + (-3)^#] &] (* _Michael De Vlieger_, May 09 2017 *)
%o A286348 (Magma) [n: n in [0..250] | IsPrime(4^n+(-3)^n)];
%o A286348 (PARI) is(n)=ispseudoprime(4^n+(-3)^n) \\ _Charles R Greathouse IV_, Jun 13 2017
%Y A286348 Cf. A059801, A081505, A283653, A285929.
%K A286348 nonn,more
%O A286348 1,2
%A A286348 _Juri-Stepan Gerasimov_, May 07 2017