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k^j + (k+1)^j is prime only for j = power of 2.

Subset of A128780 which is a subset of A068501.

Primes of the form x^4 + y^4 such that q = x^2 + y^2 and p = |y^2 - x^2| are both primes.

Primes of the form n^4 + (n+1)^4 such that q = n^2 + (n+1)^2 and p = 2n+1 are both primes; so for n in A128780.

Primes of the form x^4 + y^4 such that |y^4 - x^4| is a semiprime.

From Robert G. Wilson v, Feb 26 2017: (Start)

{q, p, a(n) = (p^2+q^2)/2}

{5, 3, 17}

{13, 5, 97}

{181, 19, 16561}

{421, 29, 89041}

{71821, 379, 2579199841}

{83641, 409, 3497992081}

{106261, 461, 5645806321}

{205441, 641, 21103207681}

{926161, 1361, 428888025121}

{1171981, 1531, 686770904161}

(End)

For k = 6 and 9, (n+1)^k + (-n)^k is always composite (i.e. (n+1)^6 + (-n)^6 = (2*n^2+2*n+1)*(n^4+2*n^3+5*n^2+4*n+1), (n+1)^9 + (-n)^9 = (3*n^2+3*n+1)*(3*n^6+9*n^5+18*n^4+21*n^3+15*n^2+6*n+1)).