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Search: id:A057102
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| A057102 |
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Congrua (possible solutions to the congruum problem): numbers n such that there are integers x, y and z with n = x^2-y^2 = z^2-x^2. |
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13
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| 24, 96, 120, 240, 336, 384, 480, 720, 840, 960, 1320, 1344, 1536, 1920, 1944, 2016, 2184, 2520, 2880, 3360, 3696, 3840, 3960, 4896, 5280, 5376, 5544, 6144, 6240, 6840, 6864, 7680, 7776, 8064, 8736, 9240, 9360, 9720, 10080, 10296, 10920, 11520, 12144
(list; graph; listen)
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OFFSET
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1,1
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COMMENT
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Each congruum is a multiple of 24; it cannot be a square.
Numbers of the form (4(x^3y-xy^2) (where x,y are integrs and x>=y). Squares of these numbers are of the form N^4-K^2 (where N belongs to A135786 and K to A135789 or A135790). Proof uses identity: (4(x^3y-xy^2))^2=(x^2+y^2)^4-(x^4 - 6x^2 y^2 + y^4)^2 - Artur Jasinski (grafix(AT)csl.pl), Nov 29 2007
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LINKS
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Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.
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EXAMPLE
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a(9)=840 since 840=29^2-1^2=41^2-29^2 (indeed also 840=37^2-23^2=47^2-37^2)
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MATHEMATICA
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a = {}; Do[Do[w = 4x^3y - 4x y^3; If[w > 0 && w < 10000, AppendTo[a, w]], {x, y, 1000}], {y, 1, 1000}]; Union[a] - Artur Jasinski (grafix(AT)csl.pl), Nov 29 2007
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CROSSREFS
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Cf. A004431 for possible values of x in definition. Cf. A057103, A055096 for triangles of all congrua and values of x.
Cf. A073120.
Cf. A135789, A135786.
Sequence in context: A055671 A090214 A103251 this_sequence A057103 A055669 A042122
Adjacent sequences: A057099 A057100 A057101 this_sequence A057103 A057104 A057105
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KEYWORD
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nonn
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AUTHOR
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Henry Bottomley (se16(AT)btinternet.com), Aug 02 2000
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