Primitive Pythagorean quintuples (a, b, c, d, e) sorted so that a <= b <= c <= d < e.
1, 1, 1, 1, 2, 1, 2, 2, 4, 5, 1, 1, 3, 5, 6, 1, 4, 4, 4, 7, 2, 2, 4, 5, 7, 2, 2, 3, 8, 9, 2, 4, 5, 6, 9, 1, 1, 7, 7, 10, 1, 3, 3, 9, 10, 1, 5, 5, 7, 10, 1, 2, 4, 10, 11, 2, 2, 7, 8, 11, 4, 4, 5, 8, 11, 1, 2, 8, 10, 13, 2, 4, 7, 10, 13, 4, 4, 4, 11, 13, 4, 5, 8, 8, 13
1
Quintuples such that e^2 = d^2 + c^2 + b^2 + a^2. There are only a finite number of cases that sum to a given e^2. It appears that e can be any positive integer except 1, 3, and multiples of 4.
T. D. Noe, Plot of 1017 quintuples
T. D. Noe, Table of 1017 quintuples
(Mma) a =.; b =.; c =.; d =.; t = {}; terms = 0; e = 0; While[terms < 100, e++; sol = Solve[a^2 + b^2 + c^2 + d^2 == e^2 && 0 < a <= b <= c <= d < e, {a, b, c, d}, Integers]; If[Length[sol] > 0, t8 = Table[{sol[[i]][[1, 2]], sol[[i]][[2, 2]], sol[[i]][[3, 2]], sol[[i]][[4, 2]]}, {i, Length[sol]}]; sol = Select[t8, GCD @@ # == 1 &]; If[Length[sol] > 0, sol = Transpose[Join[Transpose[sol], {Table[e, {Length[sol]}]}]]; t = Join[t, sol]; terms = terms + Length[sol]]]]; t
nonn,tabl
T. D. Noe, Aug 09 2017