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