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Search: id:A103256
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| A103256 |
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Positive integers x such that there exist positive integers y and z satisfying x^3 + y^3 = z^4. |
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+0
1
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| 2, 9, 18, 28, 32, 65, 70, 84, 105, 126, 144, 162
(list; graph; listen)
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OFFSET
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1,1
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COMMENT
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There are no solutions with (x,y,z) relatively prime. [Bruin]
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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.
Nils Bruin, On powers as sums of two cubes, in Algorithmic number theory (Leiden, 2000), 169-184, Lecture Notes in Comput. Sci., 1838, Springer, Berlin, 2000.
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LINKS
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Wikipedia, Beal's conjecture
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FORMULA
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A parametrized solution: (a (a^m+b^m))^m + (b(a^m+b^m))^m = (a^m+b^m)^(m+1) [From Wikipedia article - set m=3] - James McLaughlin, Jan 28 2007
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EXAMPLE
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x=9, y=18, 9^3 + 18^3 = 9^4, so 9 and 18 are on the list.
Other solutions are (2, 2, 2), (9, 18, 9), (28, 84, 28), (32, 32, 16), ...
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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 and IsSquare(Abs(P[2]))} ]; (from Geoff Bailey)
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CROSSREFS
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Sequence in context: A083419 A126082 A083707 this_sequence A028881 A083708 A037421
Adjacent sequences: A103253 A103254 A103255 this_sequence A103257 A103258 A103259
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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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Corrected and extended by Geoff Bailey (geoff(AT)maths.usyd.edu.au) using MAGMA, Jan 28 2007.
a(9)-a(12) from Jonathan Vos Post (jvospost2(AT)yahoo.com), May 27 2007
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