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Integer areas of integer-sided triangles where two sides are of prime length.
3

%I #22 Mar 08 2017 08:19:56

%S 6,12,30,60,66,72,114,120,180,210,240,330,336,360,396,420,456,660,756,

%T 780,840,900,984,1116,1200,1248,1260,1290,1320,1584,1590,1680,1710,

%U 1716,1770,1800,1980,2100,2310,2400,2460,2496,2520,2604,2640,2940,2970,3060,3120

%N Integer areas of integer-sided triangles where two sides are of prime length.

%C Subset of A188158. The length of the third side is an even composite number because the perimeter is always even.

%C The area of the triangles (a,b,c) are given by Heron's formula A = sqrt(s(s-a)(s-b)(s-c)) where its side lengths are a, b, c and semiperimeter s = (a+b+c)/2.

%C The following table gives the first values (A, a, b, c):

%C ***********************

%C * A * a * b * c *

%C ***********************

%C * 6 * 3 * 4 * 5 *

%C * 12 * 5 * 5 * 6 *

%C * 12 * 5 * 5 * 8 *

%C * 30 * 5 * 12 * 13 *

%C * 60 * 10 * 13 * 13 *

%C * 66 * 11 * 13 * 20 *

%C * 72 * 5 * 29 * 30 *

%C * 114 * 19 * 20 * 37 *

%C * 120 * 16 * 17 * 17 *

%C * 120 * 17 * 17 * 30 *

%C * 180 * 13 * 30 * 37 *

%C ....................

%H Eric W. Weisstein, <a href="http://mathworld.wolfram.com/HeronsFormula.html">Heron's Formula</a>

%e 114 is in the sequence because the triangle (19, 20, 37) => semiperimeter s = (19+20+37)/2 = 38, and A = sqrt(38*(38-19)*(38-20)*(38-37)) = 114, with 19 and 37 prime numbers.

%t area[a_, b_, c_] := Module[{s = (a + b + c)/2, a2}, a2 = s (s - a) (s - b) (s - c); If[a2 < 0, 0, Sqrt[a2]]]; goodQ[a_, b_, c_] := Module[{ar = area[a, b, c]}, ar > 0 && IntegerQ[ar]]; nn = 80; t = {}; ps = Prime[Range[2, nn]]; mx = 3*ps[[-1]]; Do[If[p <= q && goodQ[p, q, e], aa = area[p, q, e]; If[aa <= mx, AppendTo[t, aa]]], {p, ps}, {q, ps}, {e, q - p + 2, p + q - 2, 2}]; t = Union[t] (* _T. D. Noe_, Oct 01 2013 *)

%Y Cf. A188158, A229159.

%K nonn

%O 1,1

%A _Michel Lagneau_, Sep 28 2013

%E Extended by _T. D. Noe_, Sep 30 2013

%E Missing term 2970 from _Giovanni Resta_, Mar 08 2017