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A283564
Positive integers k such that k = a/(b+c) + b/(a+c) + c/(a+b) for some positive integers a, b and c where the corresponding elliptic curve has rank=1.
4
4, 6, 10, 12, 14, 16, 18, 24, 28, 32, 38, 42, 46, 48, 58, 60, 66, 76, 82, 92, 102, 112, 116, 126, 130, 132, 136, 146, 156, 158, 162, 178, 182, 184, 186, 196, 198, 200, 206, 218, 232, 266, 270, 276, 282, 304, 310, 312, 314, 318, 332, 336, 338, 346, 348, 362, 364, 378, 382, 388, 402, 408
OFFSET
1,1
COMMENTS
There are no odd numbers in this sequence.
The values for a, b and c are very large. The smallest known solutions contain 81 digits (for k=4).
The program by Jinyuan Wang gives the smallest possible solutions for all k<=14. But for k = 16 and f(16, -676, 15652) we get solutions containing 21349 digits. Emil Vlasák discovered a much shorter solution that has only 412 digits. We get it from f(16, -43928/81, -10230056/729). - Vaclav Kotesovec, Jan 22 2024
LINKS
Alon Amit, How do you find the positive integer solutions to ...?, Quora, Aug 07, 2017 [Broken link]
Andrew Bremner and Allan Macleod, An Unusual Cubic Representation Problem, Annales Mathematicae et Informaticae, volume 43 (2014), pages 29-41, see Table 2 page 38.
PROG
(Magma)
is_A283564 := function(k)
E := EllipticCurve([0, 4*k^2 + 12*k - 3, 0, 32*(k+3), 0]);
return ((Rank(E) eq 1) and (Min([g[1] : g in Generators(E)]) lt 0));
end function;
[k : k in [1..200] | is_A283564(k)]; // Robin Visser, Feb 04 2024
(Sage)
def is_A283564(k):
E = EllipticCurve([0, 4*k^2 + 12*k - 3, 0, 32*(k+3), 0])
return ((E.rank()==1) and (min([g.xy()[0] for g in E.gens()]) < 0))
print([k for k in range(1, 70) if is_A283564(k)]) # Robin Visser, Feb 04 2024
CROSSREFS
Cf. A369896.
Sequence in context: A163164 A137230 A369896 * A348005 A181794 A199536
KEYWORD
nonn
AUTHOR
Dmitry Kamenetsky, Mar 11 2017
EXTENSIONS
Definition clarified by Jimmy Gustafsson, May 08 2019
More terms from Robin Visser, Feb 04 2024
STATUS
approved