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Search: id:A103254
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| A103254 |
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Positive integers x such that there exist positive integers y and z satisfying x^3 + y^3 = z^2. |
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+0
3
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| 1, 2, 4, 7, 8, 9, 10, 11, 14, 16, 18, 21, 22, 23, 25, 26, 28, 32, 33, 34, 35, 36, 37, 38, 40, 44, 46, 49, 50, 56, 57, 63, 64, 65, 70, 72, 78, 81, 84, 86, 88, 90, 91, 92, 95, 98, 99, 100
(list; graph; listen)
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OFFSET
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1,2
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COMMENT
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A001105 is a subset (excluding 0), since (x, y, z)=(A001105(k), A001105(k), A033430(k)) satisfies x^3+y^3=z^2. - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Sep 11 2006
A parametric solution: {x,y,z} = {g*(4*e + g)*(4*e^2 + 8*e*g + g^2), 2*g*(4*e + g)*(-2*e^2 +2*e*g + g^2), 3*g^2*(4*e + g)^2*(4*e^2 + 2*e*g + g^2)}, provided (-2*e^2 +2*e*g + g^2)>0. - James McLaughlin, Jan 27 2007
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REFERENCES
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F. Beukers, The Diophantine equation Ax^p+By^q=Cz^r, Duke Math. J. 91 (1998), 61-88.
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EXAMPLE
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x=7, y=21, 7^3 + 21^3 = 98^2. 7 is the 4-th entry in the list.
Other solutions are (x, y, z)=(1, 2, 3), (4, 8, 24), (7, 21, 98), (9, 18, 81), (10, 65, 525), (11, 37, 228), (14, 70, 588), (16, 32, 192), (21, 7, 98), (22, 26, 168), (23, 1177, 40380), ...
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PROGRAM
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(MAGMA) [ k : k in [1..100] | exists{P : P in IntegralPoints(EllipticCurve([0, k^3])) | P[1] gt 0 and P[2] ne 0 } ]; (from Geoff Bailey)
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CROSSREFS
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See A103255 for another version.
Adjacent sequences: A103251 A103252 A103253 this_sequence A103255 A103256 A103257
Sequence in context: A094541 A092577 A009282 this_sequence A083454 A047542 A121619
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KEYWORD
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nonn
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AUTHOR
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Cino Hilliard (hillcino368(AT)gmail.com), Mar 20 2005
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EXTENSIONS
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Recomputed and extended to 48 terms by Geoff Bailey (geoff(AT)maths.usyd.edu.au) using MAGMA, Jan 28 2007
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