|
Search: id:A103254
|
|
|
| A103254 |
|
Positive integers x such that there exist positive integers y and z satisfying x^3 + y^3 = z^2. |
|
+0
3
|
|
| 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)
|
|
|
OFFSET
|
1,2
|
|
|
COMMENT
|
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
|
|
REFERENCES
|
F. Beukers, The Diophantine equation Ax^p+By^q=Cz^r, Duke Math. J. 91 (1998), 61-88.
|
|
EXAMPLE
|
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), ...
|
|
PROGRAM
|
(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)
|
|
CROSSREFS
|
See A103255 for another version.
Sequence in context: A092577 A009282 A141745 this_sequence A083454 A047542 A121619
Adjacent sequences: A103251 A103252 A103253 this_sequence A103255 A103256 A103257
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
Cino Hilliard (hillcino368(AT)gmail.com), Mar 20 2005
|
|
EXTENSIONS
|
Recomputed and extended to 48 terms by Geoff Bailey (geoff(AT)maths.usyd.edu.au) using MAGMA, Jan 28 2007
|
|
|
Search completed in 0.002 seconds
|
