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

0, 6, 6, 42, 66, 24, 36, 114, 966, 60, 930, 114, 126, 1290, 4230, 90, 1770, 330, 2814, 14910, 216, 4740, 1494, 420, 420, 510, 6180, 4494, 840, 570, 8382, 11790, 630, 9174, 210, 4530, 840, 2934, 45090, 3276, 22554, 1260, 24066, 336, 1386, 16716, 26586, 52182

Offset: 1

Author

Michel Lagneau, Sep 17 2013

Keywords

• nonn

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 |

Crossrefs

Cf. A226453.

Programs

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: