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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, 271, 294, 307, 342, 368, 416, 455, 511, 567, 602, 608, 670, 728, 762, 948, 999, 1008, 1090, 1116, 1183, 1281, 1330, 1442, 1455, 1468, 1727, 1736, 1776, 1792, 1935, 1984, 2015, 2054, 2106, 2119, 2190, 2196, 2246, 2439
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OFFSET
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1,1
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COMMENTS
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Warning: these entries have not been proved to be correct! There may be missing terms. - N. J. A. Sloane, 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 Mc Laughlin, Jan 27 2007
Conjecture is false: 126^4 = 639^3 - 207^3, the largest prime divisor of 207 is 23 and neither 126 nor 639 is divisible by 23. For max(x,y) < 4.3*10^9, there are no additional terms < 2440. It is likely this is true for all x,y. - Chai Wah Wu, Jan 17 2016
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LINKS
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EXAMPLE
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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), ...
813^4 = 7588^3 - 271^3, 614^4 = 5219^3 - 307^3, 903^4 = 8729^3 - 602^3, 6162^4 = 112970^3 - 2054^3, 4492^4 = 74118^3 - 2246^3. - Chai Wah Wu, Jan 15 2016
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MATHEMATICA
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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] (* Jean-François Alcover, Sep 06 2011 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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Terms 271, 307, 602, 2054 and 2246 added by Chai Wah Wu, Jan 15 2016
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STATUS
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approved
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