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A103267
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Positive integers x such that there exist positive integers y and z satisfying y^3 - x^3 = z^4.
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0
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7, 26, 38, 63, 111, 112, 124, 207, 215, 234, 244, 294, 342, 368, 416, 455, 511, 567, 608, 670, 728, 762, 948, 999, 1008, 1090, 1116, 1183, 1281, 1330, 1442, 1455, 1468, 1727, 1736, 1776, 1792, 1935, 1984, 2015, 2106, 2119, 2190, 2196, 2439
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,1
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COMMENTS
| Warning: these entries have not been proved to be correct! There may be missing terms. - N. J. A. Sloane (njas(AT)research.att.com), Jan 05 2007
Conjecture: The greatest prime divisor of x divides y and z.
Many of these values are derived from the essentially trivial solution {x,y,z} ={(a + b)*b*(3*a^2 + 3*a*b + b^2), a*b*(3*a^2 + 3*a*b + b^2), b*(3*a^2 + 3*a*b + b^2)}. This solution follows from the fact that (a+b)^3-a^3 = b*(3*a^2 + 3*a*b + b^2) and that multiplying this equation by [b*(3*a^2 + 3*a*b + b^2)]^3 gives a solution to y^3 - x^3 = z^4. - James McLaughlin, Jan 27 2007
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REFERENCES
| F. Beukers, The Diophantine equation Ax^p+By^q=Cz^r, Duke Math. J. 91 (1998), 61-88.
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EXAMPLE
| x=762, y=889, 889^3 - 762^3 = 127^4, so 762 is on the list.
Other solutions: (x,y,z) = (26, 26, 78), (38, 19, 57), (63, 63, 252), (111, 37, 148), (112, 56, 224), (124, 124, 620), (207, 126, 639), (215, 215, 1920), (234, 117, 585), (244, 61, 305), (294, 98, 490), (342, 342, 2394), (368, 161, 897), (416, 208, 1248), (455, 91, 546), (567, 189, 1134), (608, 152, 912), (670, 335, 2345), (762, 127, 889), (948, 316, 2212), (1090, 218, 1526), (1116, 279, 1953), (1183, 169, 1352), (1736, 217, 1953), (1776, 296, 2368), (2119, 273, 2470), (2439, 271, 2710), ...
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MATHEMATICA
| r[x_, z_] := Reduce[y > 0 && z > 0 && y^3 - x^3 == z^4, y, Integers]; ok[x_] := ok[x] = Length[Union[Table[r[x, z], {z, 1, x}]]] > 1; Select[Range[2500], ok] (* From Jean-François Alcover, Sep 06 2011 *)
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CROSSREFS
| Sequence in context: A098127 A131905 A110927 * A125972 A063153 A063578
Adjacent sequences: A103264 A103265 A103266 * A103268 A103269 A103270
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KEYWORD
| nonn
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AUTHOR
| Cino Hilliard (hillcino368(AT)gmail.com), Mar 20 2005
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EXTENSIONS
| Missing terms inserted by Jean-François Alcover (jf.alcover(AT)gmail.com), Sep 06 2011
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