## S001062

Primitive Pythagorean quadruples (a, b, c, d) sorted so that a <= b <= c < d.

1, 2, 2, 3, 2, 3, 6, 7, 1, 4, 8, 9, 4, 4, 7, 9, 2, 6, 9, 11, 6, 6, 7, 11, 3, 4, 12, 13, 2, 5, 14, 15, 2, 10, 11, 15, 1, 12, 12, 17, 8, 9, 12, 17, 1, 6, 18, 19, 6, 6, 17, 19, 6, 10, 15, 19, 4, 5, 20, 21, 4, 8, 19, 21, 4, 13, 16, 21, 8, 11, 16, 21, 3, 6, 22, 23

1

Quadruples such that d^2 = c^2 + b^2 + a^2. There are only a finite number of cases that sum to a given d^2. Only odd values of d appear. The quadruples are reversed in S001061. The Mathematica program is very different from the one in S001061.

T. D. Noe, Plot of 1018 quadruples

T. D. Noe, Table of 1018 quadruples

Eric W. Weisstein, MathWorld: Pythagorean Quadruple

Wikipedia, Pythagorean quadruple

(Mma) a =.; b =.; c =.; t = {}; terms = 0; d = -1; While[terms < 100, d = d + 2; sol = Solve[a^2 + b^2 + c^2 == d^2 && 0 < a <= b <= c <= d, {a, b, c}, Integers]; If[Length[sol] > 0, t8 = Table[{sol[[i]][[1, 2]], sol[[i]][[2, 2]], sol[[i]][[3, 2]]}, {i, Length[sol]}]; sol = Select[t8, GCD @@ # == 1 &]; sol = Transpose[Join[Transpose[sol], {Table[d, {Length[sol]}]}]]; t = Join[t, sol]; terms = terms + Length[sol]]]; t

Cf.S001069.

nonn,tabl

T. D. Noe, Aug 09 2017

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