A293694 Numbers z such that x^2 + y^8 = z^2 for positive integers x and y.

20, 34, 65, 135, 320, 369, 544, 1040, 1095, 1305, 1350, 1404, 1620, 1625, 1746, 1971, 2056, 2160, 2379, 2754, 3060, 3281, 3996, 4100, 4470, 5120, 5265, 5904, 6625, 7825, 7830, 8194, 8575, 8704, 8796, 10250, 10935, 11125, 11700, 12500, 13154, 14500, 15579

Offset: 1

Author

XU Pingya, Oct 16 2017

Keywords

• nonn

Comments

Let i, j and k be nonnegative integers, m > n be positive integers. As ((m^2 - n^2)^(4*i+1) * (2*m*n)^(4*j+3) * (m^2 + n^2)^(4*k))^2 + ((m^2 - n^2)^i * (2*m*n)^(j+1) * (m^2 + n^2)^k)^8 = ((m^2 - n^2)^(4*i) * (2*m*n)^(4*j+3) * (m^2 + n^2)^(4*k+1))^2, so that the number of the form (m^2 - n^2)^(4*i) * (2*m*n)^(4*j+3) * (m^2 + n^2)^(4*k+1) is a term.

When (x, y, z) is a solution of x^2 + y^4 = z^2 (i.e., z = A271576(n)), (x^(4*i+1) * y^(4*j+2) * z^(4*k), x^i * y^(j+1) * z^k, x^(4*i) * y^(4*j+2) * z^(4*k+1)) is a solution of x^2 + y^8 = z^2.

When (x, y, z) is a solution of x^2 + y^6 = z^2 (i.e., z = A293690(n)), (x^(4*i+1) * y^(4*j+1) * z^(4*k), x^i * y^(j+1) * z^k, x^(4*i) * y^(4*j+1) * z^(4*k+1)) is a solution of x^2 + y^8 = z^2.

When (x, y, z) is a solution of x^2 + y^8 = z^2, (x^(4*i+1) * y^(4*j) * z^(4*k), x^i * y^(j+1) * z^k, x^(4*i) * y^(4*j) * z^(4*k+1)) is also a solution of x^2 + y^8 = z^2.

Examples

			12^2 + 2^8 = 20^2, 20 is a term.
63^2 + 2^8 = 65^2, 65 is a term.
		

Links

• Karl-Heinz Hofmann, Table of n, a(n) for n = 1..13695

Crossrefs

Cf. A009003, A228946, A271576, A293283, A293690, A293692.

Programs

Mathematica

z[n_] := Count[n^2 - Range[(n^2 - 1)^(1/8)]^8, _?(IntegerQ[Sqrt[#]] &)] > 0; Select[Range[16000], z]