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A103267 Positive integers x such that there exist positive integers y and z satisfying y^3 - x^3 = z^4. 0
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

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 McLaughlin, 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

LINKS

Table of n, a(n) for n=1..50.

F. Beukers, The Diophantine equation Ax^p+By^q=Cz^r, Duke Math. J. 91 (1998), 61-88.

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), ...

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

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] (* Jean-François Alcover, Sep 06 2011 *)

CROSSREFS

Sequence in context: A283427 A131905 A110927 * A125972 A063153 A063578

Adjacent sequences:  A103264 A103265 A103266 * A103268 A103269 A103270

KEYWORD

nonn

AUTHOR

Cino Hilliard, Mar 20 2005

EXTENSIONS

Missing terms inserted by Jean-François Alcover, Sep 06 2011

Terms 271, 307, 602, 2054 and 2246 added by Chai Wah Wu, Jan 15 2016

STATUS

approved

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Last modified August 15 12:26 EDT 2020. Contains 336496 sequences. (Running on oeis4.)