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Numbers z such that x^2 + y^6 = z^2 for positive integers x and y.
2

%I #5 Oct 15 2017 01:00:51

%S 10,17,45,80,123,136,225,234,260,270,291,325,360,365,459,510,514,640,

%T 666,745,984,1025,1088,1215,1225,1250,1305,1450,1466,1565,1740,1753,

%U 1800,1872,1950,1970,2022,2080,2125,2160,2328,2600,2628,2880,2920,3172,3185

%N Numbers z such that x^2 + y^6 = z^2 for positive integers x and y.

%C Let i, j and k are nonegtive integers, m > n are positive integers. As [(m^2 - n^2)^(3i+1) * (2mn)^(3j + 2) * (m^2 + n^2)^(3k)]^2 + [(m^2 - n^2)^i * (2mn)^(j + 1) * (m^2 + n^2)^k]^6 = [(m^2 - n^2)^(3i) * (2mn)^(3j + 2) * (m^2 + n^2)^(3k+1)]^2, so that number of form (m^2 - n^2)^(3i) * (2mn)^(3j + 2) * (m^2 + n^2)^(3k+1) is a term in sequence.

%C When (x, y, z) is solution of x^2 + y^4 = z^2 (i.e., z = A271576(n)), (x^(3i+1) * y^(3j + 1) * z^(3k), (x^i * y^(j + 1) * z^k)^6, x^(3i) * y^(3j + 1) * z^(3k+1) is solution of x^2 + y^6 = z^2.

%C When (x, y, z) is solution of x^2 + y^6 = z^2, (x^(3i+1) * y^(3j) * z^(3k), x^(3i) * y^(j + 1) * z^k, x^(3i) * y^(3j) * z^(3k+1)) is also.

%e 6^2 + 2^6 = 10^2, 10 is a term.

%e 15^2 + 2^6 = 17^2, 17 is a term.

%t z[n_] := Count[n^2 - Range[(n^2 - 1)^(1/6)]^6, _?(IntegerQ[Sqrt[#]] &)] > 0; Select[Range[3200], z]

%Y Cf. A009003, A070745, A228946, A271576, A274035.

%K nonn

%O 1,1

%A _XU Pingya_, Oct 14 2017