Integer triangles with 120° angle

 

Law of cosines
https://en.wikipedia.org/wiki/Law_of_cosines

c^2 \; = \; a^2 \; + \; b^2 \; - \; 2 \, a \, b \, \cos \, \gamma

If   \gamma \; = \; 120^\circ   —>   \cos 120^\circ \; = \; -0.5

then,     c^2 \; = \; a^2 \; + \; b^2 \; + \; a \, b

 
 
To find triples   (a, b, c)   that satisfy the relation   c^2 \; = \; a^2 \; + \; b^2 \; + \; a \, b

We use the parametric equations :

a \; = \; m^2 \; - \; n^2
b \; = \; 2 \, m \, n \; + \; n^2
c \; = \; m^2 \; + \; m \, n \; + \; n^2

where   m   and   n   are integers and   m \; > \; n

(m^2 - n^2 )^2 \; + \; (m^2 - n^2 ) \,(2 \, m \, n + n^2 ) \; + \; (2 \, m \, n + n^2 )^2
= \; (m^2 + m \, n + n^2 )^2

Perimeter   P :

P \; = \; m^2 \; - \; n^2 \; + \; 2 \, m \, n \; + \; n^2 \; + \; m^2 \; + \; m \, n \; + \; n^2
P \; = \; 2 \, m^2 \; + \; 3 \, m \, n \; + \; n^2
P \; = \; (2 \, m \; + \; n) \, (m \; + \; n)

 

The first few primitive triples are:

(3, \; 5, \; 7)
(5, \; 16, \; 19)
(7, \; 8, \; 13)
(7, \; 33, \; 37)
(9, \; 56, \; 61)
(11, \; 24, \; 31)
(11, \; 85, \; 91)
(13, \; 35, \; 43)
(17, \; 63, \; 73)
(19, \; 80, \; 91)

 
In the case where   (a, \; b)   are consecutive integers:

The first few triples such that   (a, \; a+1, \; c)

(7, \; 8, \; 13)
(104, \; 105, \; 181)
(1455, \; 1456, \; 2521)
(20272, 20273, \; 35113)
(282359, \; 282360, \; 489061)
(3932760, \; 3932761, \; 6811741)
(54776287, \; 54776288, \; 94875313)
(762935264, \; 762935265, \; 1321442641)
(10626317415, \; 10626317416, \; 18405321661)
(148005508552, \; 148005508553, \; 256353060613)
(2061450802319, \; 2061450802320, \; 3570537526921)
(28712305723920, \; 28712305723921, \; 49731172316281)
(399910829332567, \; 399910829332568, \; 692665874901013)
(5570039304932024, \; 5570039304932025, \; 9647591076297901)

 
The first few triples such that   (a, \; a+2, \; c)

(3, \; 5, \; 7)
(14, \; 16, \; 26)
(55, \; 57, \; 97)
(208, \; 210, \; 362)
(779, \; 781, \; 1351)
(2910, \; 2912, \; 5042)
(10863, \; 10865, \; 18817)
(40544, \; 40546, \; 70226)
(151315, \; 151317, \; 262087)
(564718, \; 564720, \; 978122)
(2107559, \; 2107561, \; 3650401)
(7865520, \; 7865522, \; 13623482)
(29354523, \; 29354525, \; 50843527)
(109552574, \; 109552576, \; 189750626)

 
The first few triples such that   (a, \; a+3, \; c)

(21, \; 24, \; 39)
(312, \; 315, \; 543)
(4365, \; 4368, \; 7563)
(60816, \; 60819, \; 105339)
(847077, \; 847080, \; 1467183)
(11798280, \; 11798283, \; 20435223)
(164328861, \; 164328864, \; 284625939)
(2288805792, \; 2288805795, \; 3964327923)
(31878952245, \; 31878952248, \; 55215964983)
(444016525656, \; 444016525658, \; 769059181839)
(6184352406957, \; 6184352406960, \; 10711612580763)
(86136917171760, \; 86136917171763, \; 149193516948843)
(1199732487997701, \; 1199732487997704, \; 2077997624703039)
(16710117914796072, \; 16710117914796075, \; 28942773228893703)

 

 
 
 
 
 
 
 
 
 
 
 
 
 
 

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About benvitalis

math grad - Interest: Number theory
This entry was posted in Number Puzzles and tagged , . Bookmark the permalink.

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