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Search: id:A156687
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| A156687 |
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Perimeters of Pythagorean triangles that can be constructed in exactly 5 different ways. |
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+0
1
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| 420, 660, 924, 1008, 1080, 1200, 1512, 1584, 1716, 1800, 1872, 1890, 2700, 3150, 3168, 3240, 3480, 3528, 3570, 3720, 3744, 4410, 4440, 4536, 4590, 4704, 4872, 4896, 4950, 5208, 5292, 5472, 5600, 5670, 6000, 6090, 6210, 6216, 6624, 6630, 6660, 6888
(list; graph; listen)
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OFFSET
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1,1
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COMMENT
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For any given N we can always find at least N Pythagorean triangles with the same perimeter.
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REFERENCES
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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.
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LINKS
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Ron Knott, Right-angled Triangles and Pythagoras' Theorem
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EXAMPLE
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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.
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MATHEMATICA
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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 &]
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CROSSREFS
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A099831, A099832, A099833, A009129, A010814
Sequence in context: A069064 A024410 A070237 this_sequence A147775 A135196 A145678
Adjacent sequences: A156684 A156685 A156686 this_sequence A156688 A156689 A156690
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KEYWORD
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easy,nice,nonn
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AUTHOR
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Ant King (mathstutoring(AT)ntlworld.com), Feb 18 2009
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