## S001066

Primitive Pythagorean octuples (a, b, c, d, e, f, g, h) sorted.

1, 1, 1, 1, 2, 2, 2, 4, 1, 1, 1, 1, 1, 2, 4, 5, 1, 2, 2, 2, 2, 2, 2, 5, 1, 1, 1, 2, 2, 3, 4, 6, 1, 1, 1, 1, 2, 4, 5, 7, 1, 1, 2, 3, 3, 3, 4, 7, 1, 2, 2, 2, 2, 4, 4, 7, 2, 2, 2, 2, 2, 2, 5, 7, 1, 1, 1, 2, 2, 2, 7, 8, 1, 1, 1, 2, 4, 4, 5, 8, 1, 1, 2, 2, 2, 5, 5, 8

1

Octuples such that h^2 = g^2 + f^2 + e^2 + d^2 + c^2 + b^2 + a^2. There are only a finite number of cases that sum to a given h^2.

T. D. Noe, Plot of 1146 octuples

T. D. Noe, Table of 1146 octuples

(Mma) a =.; b =.; c =.; d =.; e =.; f =.; g =.; t = {}; terms = 0; h = 0; While[terms < 100, h++; sol = Solve[a^2 + b^2 + c^2 + d^2 + e^2 + f^2 + g^2 == h^2 && 0 < a <= b <= c <= d <= e <= f <= g <= h, {a, b, c, d, e, f, g}, Integers]; If[Length[sol] > 0, t8 = Table[Table[sol[[i]][[j, 2]], {j, 7}], {i, Length[sol]}]; sol = Select[t8, GCD @@ # == 1 &]; If[Length[sol] > 0, sol = Transpose[Join[Transpose[sol], {Table[h, {Length[sol]}]}]]; t = Join[t, Sort[sol]]; terms = terms + Length[sol]]]]; t

Cf.S001069.

nonn,tabl

T. D. Noe, Aug 10 2017