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

0, 3, 4, 7, 16, 17, 59, 283, 311, 383, 499, 521, 541, 599, 1193, 1993, 2671, 7547, 24019, 46301, 48121, 68597, 91283, 131497, 148663, 184463, 341233

Offset: 1

Author

Juri-Stepan Gerasimov, May 07 2017

Keywords

• nonn
• more

Comments

Numbers n such that (1 + k)^n + (-k)^n is prime:

0 (k = 0);

A285929 (k = 1);

A283653 (k = 2);

this sequence (k = 3);

0, 2, 3, 4, 43, 59, 191, 223, ... (k = 4);

0, 2, 5, 8, 11, 13, 16, 23, 61, 83, ...(k = 5);

0, 3, 4, 7, 16, 29, 41, 67, ... (k = 6);

0, 2, 7, 11, 16, 17, 29, 31, 79, 43, 131, 139, ... (k = 7);

0, 4, 7, 29, 31, 32, 67, ... (k = 8);

0, 2, 3, 4, 7, 11, 19, 29, ... (k = 9);

0, 3, 5, 19, 32, ... (k = 10);

0, 3, 7, 89, 101, ... (k = 11);

0, 2, 4, 17, 31, 32, 41, 47, 109, 163, ... (k = 12);

0, 3, 4, 11, 83, ... (k = 13);

0, 2, 3, 4, 16, 43, 173, 193, ... (k = 14);

0, 43, ... (k = 15);

0, 4, 5, 7, 79, ... (k = 16);

0, 2, 3, 8, 13, 71, ... (k = 17);

0, 1607, ... (k = 18);

...

Primes of the form (1 + n)^(2^n) + n: 5, 83, 65539, 7958661109946400884391941, ...

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, ...

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.

Examples

			3 is in this sequence because 4^3 + (-3)^3 = 37 is prime.
4 is in this sequence because 4^4 + (-3)^4 = 337 is prime.
		

Crossrefs

Cf. A059801, A081505, A283653, A285929.

Programs

Magma

[n: n in [0..250] | IsPrime(4^n+(-3)^n)];

Mathematica

Select[Range[0, 3000], PrimeQ[4^# + (-3)^#] &] (* Michael De Vlieger, May 09 2017 *)

PARI

is(n)=ispseudoprime(4^n+(-3)^n) \\ Charles R Greathouse IV, Jun 13 2017