OFFSET
1,2
COMMENTS
Conjecture: for all prime p > 2 there exists an integer-sided triangle with integer area where at least one side is of length p.
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, ...
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:
(3, 4, 5) => 12 = 4*3;
(11, 13, 20) => 44 = 4*11;
(19, 20, 37) => 76 = 4*19;
(43, 61, 68) => 172 = 4*43;
(59, 68, 109) => 236 = 4*59;
(131, 181, 212) => 524 = 4*131;
(139, 157, 260) => 556 = 4*139;
(179, 260, 277) => 716 = 4*179.
The first 25 values (prime(n), smallest area, a, b, c) are:
+---------+-------+-----+-----+-----+
| prime(n)| Area | a | b | c |
+---------+-------+-----+-----+-----+
| 2 | 0 | 0 | 0 | 0 |
| 3 | 6 | 3 | 4 | 5 |
| 5 | 6 | 3 | 4 | 5 |
| 7 | 42 | 7 | 15 | 20 |
| 11 | 66 | 11 | 13 | 20 |
| 13 | 24 | 4 | 13 | 15 |
| 17 | 36 | 9 | 10 | 17 |
| 19 | 114 | 19 | 20 | 37 |
| 23 | 966 | 23 | 140 | 159 |
| 29 | 60 | 6 | 25 | 29 |
| 31 | 930 | 31 | 68 | 87 |
| 37 | 114 | 19 | 20 | 37 |
| 41 | 126 | 15 | 28 | 41 |
| 43 | 1290 | 43 | 61 | 68 |
| 47 | 4230 | 47 | 425 | 468 |
| 53 | 90 | 4 | 51 | 53 |
| 59 | 1770 | 59 | 68 | 109 |
| 61 | 330 | 11 | 60 | 61 |
| 67 | 2814 | 67 | 85 | 116 |
| 71 | 14910 | 71 | 447 | 476 |
| 73 | 216 | 9 | 73 | 80 |
| 79 | 4740 | 79 | 241 | 312 |
| 83 | 1494 | 83 | 85 | 164 |
| 89 | 420 | 41 | 50 | 89 |
| 97 | 420 | 26 | 73 | 97 |
MAPLE
with(numtheory):nn:=500: for m from 2 to 40 do: q:=ithprime(m):ii:=0:for a from 1
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):
:else fi:od:od:od:od:
CROSSREFS
KEYWORD
nonn
AUTHOR
Michel Lagneau, Sep 17 2013
STATUS
approved