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A229159 Smallest integer areas of integer-sided triangles where at least one side is of length prime(n). 2
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 (list; graph; refs; listen; history; text; internal format)
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 |
LINKS
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
Cf. A226453.
Sequence in context: A279535 A125510 A117859 * A102901 A014435 A175550
KEYWORD
nonn
AUTHOR
Michel Lagneau, Sep 17 2013
STATUS
approved

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Last modified May 20 03:57 EDT 2024. Contains 372703 sequences. (Running on oeis4.)