OFFSET
1,1
COMMENTS
For any given N we can always find at least N Pythagorean triangles with the same perimeter.
REFERENCES
Sierpinski, W.; Pythagorean Triangles, Dover Publications, Inc., Mineola, New York, 2003.
Beiler, Albert H.; Recreations In The Theory Of Numbers, Chapter XIV, The Eternal Triangle, Dover Publications Inc., New York, 1964, pp. 104-134.
LINKS
Ray Chandler, Table of n, a(n) for n = 1..10000
EXAMPLE
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.
MATHEMATICA
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 &]
CROSSREFS
KEYWORD
easy,nice,nonn
AUTHOR
Ant King, Feb 18 2009
STATUS
approved