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%I #7 Feb 07 2020 14:23:28
%S 420,660,924,1008,1080,1200,1512,1584,1716,1800,1872,1890,2700,3150,
%T 3168,3240,3480,3528,3570,3720,3744,4410,4440,4536,4590,4704,4872,
%U 4896,4950,5208,5292,5472,5600,5670,6000,6090,6210,6216,6624,6630,6660,6888
%N Perimeters of Pythagorean triangles that can be constructed in exactly 5 different ways.
%C For any given N we can always find at least N Pythagorean triangles with the same perimeter.
%D Sierpinski, W.; Pythagorean Triangles, Dover Publications, Inc., Mineola, New York, 2003.
%D Beiler, Albert H.; Recreations In The Theory Of Numbers, Chapter XIV, The Eternal Triangle, Dover Publications Inc., New York, 1964, pp. 104-134.
%H Ray Chandler, <a href="/A156687/b156687.txt">Table of n, a(n) for n = 1..10000</a>
%H Ron Knott, <a href="http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Pythag/pythag.html">Right-angled Triangles and Pythagoras' Theorem</a>
%e As 924 is the third smallest integer that can occur as the perimeter of exactly 5 Pythagorean triples - specifically (42,440,442), (77,420,427), (132,385,407), (198,336,390) and (231,308,385) - then a(3)=924.
%t SetSystemOptions["ReduceOptions"->{"DiscreteSolutionBound"->100000}];AllPerimeterTriples[n_Integer]/;n>0:=Module[{result=Reduce[Reduce[{x^2+y^2==z^2,z>y>x>0,Element[{x,y,z},Integers],x+y+z==n},{x,y,z}]]},If[result===False,{},Sort[{x,y,z}/.{ToRules[result]}]]];Select[Range[10000],Length[AllPerimeterTriples[ # ]]==5 &]
%Y Cf. A098714, A099831, A099832, A099833, A009129, A010814.
%K easy,nice,nonn
%O 1,1
%A _Ant King_, Feb 18 2009
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