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Integer areas of integer-sided triangles such that the length of two sides are Fibonacci numbers.
0

%I #4 Jan 28 2014 12:00:59

%S 6,12,30,60,126,396,480,840,1452,3120,3696,5040,7332,21840,47520,

%T 50460,52440,66000,70980,148512,178608,185472,483360,1021020,1025640,

%U 1493280,2299920,2772480,2985234,3357900,4442256,5362980,6990060,8233680,9801000,14058330

%N Integer areas of integer-sided triangles such that the length of two sides are Fibonacci numbers.

%C Conjecture: there exists only one triangle where three sides are of Fibonacci number length. This triangle is (5, 5, 8), and the area is A = 12.

%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 | 8 |

%C | 30 | 5 | 12 | 13 |

%C | 60 | 10 | 13 | 13 |

%C | 126 | 13 | 20 | 21 |

%C | 396 | 34 | 55 | 87 |

%C | 480 | 34 | 34 | 60 |

%C | 840 | 21 | 89 | 100 |

%C | 1452 | 55 | 55 | 66 |

%C | 3120 | 78 | 89 | 89 |

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

%p with(combinat,fibonacci):nn:=30000:n1:=40:lst1:={}:for i from 1 to n1 do:a:=fibonacci(i): for j from i to n1 do: b:=fibonacci(j):for c from 1 to nn do:p:=(a+b+c)/2:x:=p*(p-a)*(p-b)*(p-c):if x>0 then q:=sqrt(x):if q=floor(q) then lst1:=lst1 union {q}:printf ( "%d %d %d %d \n",q,a,b,c):else fi:fi:od:od:od:print(lst1):

%t n1 = 30000; n2 = 40; lst = {}; Do[s = (Fibonacci[a] + Fibonacci[b] + c)/2; If[IntegerQ[s], area2 = s (s - Fibonacci[a]) (s - Fibonacci[b]) (s - c); If[0 < area2 && IntegerQ[Sqrt[area2]], AppendTo[lst, Sqrt[area2]]]], {a, n2}, {b, a}, {c,1,n1}]; Union[lst]

%Y Cf. A000045.

%K nonn

%O 1,1

%A _Michel Lagneau_, Jan 28 2014