## S000996

Triples x >= y >= z that form the sides of a primitive triangle having integer area.

5, 4, 3, 6, 5, 5, 8, 5, 5, 13, 12, 5, 13, 13, 10, 15, 13, 4, 15, 14, 13, 17, 10, 9, 17, 15, 8, 17, 17, 16, 20, 13, 11, 20, 15, 7, 21, 17, 10, 21, 20, 13, 24, 13, 13, 25, 17, 12, 25, 24, 7, 25, 25, 14, 26, 25, 3, 26, 25, 17, 28, 25, 17, 29, 21, 20, 29, 25, 6

1

The lengths of the sides of a primitive triangle have no common factor. All these triangles have an even side and two odd sides. The shortest side can be no smaller than 3. The areas are given in S000999. If only right triangles are considered, then S000991 shows the sides. There are an infinite number of triangles whose sides have lengths s+1, s, 3. The s are given in A072221.

T. D. Noe, Plot of 1000 triples

T. D. Noe, Table of 1000 triples

(Mma) area[a_Integer, b_Integer, c_Integer] := Sqrt[(a + b + c) (-a + b + c) (a - b + c) (a + b - c)] / 4; nn = 50; t = {}; Do[If[GCD[a, b, c] == 1 && a < b + c, ar = area[a, b, c]; If[IntegerQ[ar], AppendTo[t, {a, b, c}]]], {a, nn}, {b, a}, {c, b}]; t

Cf. A072221, S000991, S000998, S000999.

nonn,tabl,nice

T. D. Noe, Apr 10 2017