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Integer areas of integer-sided triangles where the lengths of two of the sides are cubes.
0

%I #11 Dec 07 2019 00:38:58

%S 480,4200,5148,7500,30720,65520,268800,329472,349920,480000,960960,

%T 1684980,1713660,1884960,1966080,2413320,2419560,3061800,3752892,

%U 4193280,5467500,7500000,8168160,10022520,11166960,17203200,17915040,18462300,21086208,22394880,28964040

%N Integer areas of integer-sided triangles where the lengths of two of the sides are cubes.

%C Subset of A188158.

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

%C The areas of the nonprimitive triangles of sides (a*k^3, b*k^3, c*k^3), k = 1,2,... are in the sequence with the value A*k^6.

%C There may be multiple triangles with the same area (see the table of examples below).

%e The following table gives the initial values of (A, a, b, c):

%e +--------+------+-------+-------+

%e | A | a | b | c |

%e +--------+------+-------+-------+

%e | 480 | 8 | 123 | 125 |

%e | 4200 | 70 | 125 | 125 |

%e | 4200 | 125 | 125 | 240 |

%e | 5148 | 88 | 125 | 125 |

%e | 5148 | 125 | 125 | 234 |

%e | 7500 | 125 | 125 | 150 |

%e | 7500 | 125 | 125 | 200 |

%e | 30720 | 64 | 984 | 1000 |

%e | 65520 | 125 | 2088 | 2197 |

%e | 268800 | 560 | 1000 | 1000 |

%e | 268800 | 1000 | 1000 | 1920 |

%e | 329472 | 704 | 1000 | 1000 |

%e | 329472 | 1000 | 1000 | 1872 |

%e | 349920 | 216 | 3321 | 3375 |

%e .................................

%t nn=600;lst={};Do[s=(a^3+b^3+c)/2;If[IntegerQ[s],area2=s (s-a^3)(s-b^3) (s-c);If[0<area2&&IntegerQ[Sqrt[area2]],AppendTo[lst,Sqrt[area2]]]],{a,nn},{b,a},{c,1,50000}];Union[lst]

%Y Cf. A188158, A232461.

%K nonn

%O 1,1

%A _Michel Lagneau_, Nov 16 2019