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A229159 Smallest integer areas of integer-sided triangles where at least one side is of length prime(n).

%I A229159 #18 Sep 16 2017 03:44:13
%S A229159 0,6,6,42,66,24,36,114,966,60,930,114,126,1290,4230,90,1770,330,2814,
%T A229159 14910,216,4740,1494,420,420,510,6180,4494,840,570,8382,11790,630,
%U A229159 9174,210,4530,840,2934,45090,3276,22554,1260,24066,336,1386,16716,26586,52182
%N A229159 Smallest integer areas of integer-sided triangles where at least one side is of length prime(n).
%C A229159 Conjecture: for all prime p > 2 there exists an integer-sided triangle with integer area where at least one side is of length p.
%C A229159 There exist triangles of integer area and integer side lengths having two sides whose lengths are distinct prime numbers; for example, (3,4,5), (11,13,20), (19, 20,37), (43,61,68), (59,68,109), (11,60,61), (79,241, 312), (41,50,89), (26,73,97), ... corresponding to the areas 6, 66, 114, 1290, 1770, 330, 4740, 420, 420, ...
%C A229159 Observation: there exist some integer-area, integer-sided triangles with two prime sides such that the perimeter equals 4 times the smaller prime. For example:
%C A229159     (3,   4,   5) =>  12 = 4*3;
%C A229159    (11,  13,  20) =>  44 = 4*11;
%C A229159    (19,  20,  37) =>  76 = 4*19;
%C A229159    (43,  61,  68) => 172 = 4*43;
%C A229159    (59,  68, 109) => 236 = 4*59;
%C A229159   (131, 181, 212) => 524 = 4*131;
%C A229159   (139, 157, 260) => 556 = 4*139;
%C A229159   (179, 260, 277) => 716 = 4*179.
%C A229159 The first 25 values (prime(n), smallest area, a, b, c) are:
%C A229159 +---------+-------+-----+-----+-----+
%C A229159 | prime(n)|  Area |  a  |  b  |  c  |
%C A229159 +---------+-------+-----+-----+-----+
%C A229159 |     2   |     0 |   0 |   0 |   0 |
%C A229159 |     3   |     6 |   3 |   4 |   5 |
%C A229159 |     5   |     6 |   3 |   4 |   5 |
%C A229159 |     7   |    42 |   7 |  15 |  20 |
%C A229159 |    11   |    66 |  11 |  13 |  20 |
%C A229159 |    13   |    24 |   4 |  13 |  15 |
%C A229159 |    17   |    36 |   9 |  10 |  17 |
%C A229159 |    19   |   114 |  19 |  20 |  37 |
%C A229159 |    23   |   966 |  23 | 140 | 159 |
%C A229159 |    29   |    60 |   6 |  25 |  29 |
%C A229159 |    31   |   930 |  31 |  68 |  87 |
%C A229159 |    37   |   114 |  19 |  20 |  37 |
%C A229159 |    41   |   126 |  15 |  28 |  41 |
%C A229159 |    43   |  1290 |  43 |  61 |  68 |
%C A229159 |    47   |  4230 |  47 | 425 | 468 |
%C A229159 |    53   |    90 |   4 |  51 |  53 |
%C A229159 |    59   |  1770 |  59 |  68 | 109 |
%C A229159 |    61   |   330 |  11 |  60 |  61 |
%C A229159 |    67   |  2814 |  67 |  85 | 116 |
%C A229159 |    71   | 14910 |  71 | 447 | 476 |
%C A229159 |    73   |   216 |   9 |  73 |  80 |
%C A229159 |    79   |  4740 |  79 | 241 | 312 |
%C A229159 |    83   |  1494 |  83 |  85 | 164 |
%C A229159 |    89   |   420 |  41 |  50 |  89 |
%C A229159 |    97   |   420 |  26 |  73 |  97 |
%p A229159 with(numtheory):nn:=500: for m from 2 to 40 do: q:=ithprime(m):ii:=0:for a from 1
%p A229159   to nn while(ii=0) do: for b from a to nn while(ii=0) do: for c from b to nn while(ii=0) do: p:=(a+b+c)/2 : x:=p*(p-a)*(p-b)*(p-c): if x>0 then x0:= sqrt(x):else fi:if (x0=floor(x0) and a=q) or (x0=floor(x0) and b=q) or (x0=floor(x0) and c=q)then ii:=1: printf ( "%d %d %d %d %d \n",q,x0,a,b,c):
%p A229159 :else fi:od:od:od:od:
%Y A229159 Cf. A226453.
%K A229159 nonn
%O A229159 1,2
%A A229159 _Michel Lagneau_, Sep 17 2013