A283564 - OEIS

%I #53 Jun 19 2024 09:28:39

%S 4,6,10,12,14,16,18,24,28,32,38,42,46,48,58,60,66,76,82,92,102,112,

%T 116,126,130,132,136,146,156,158,162,178,182,184,186,196,198,200,206,

%U 218,232,266,270,276,282,304,310,312,314,318,332,336,338,346,348,362,364,378,382,388,402,408

%N 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.

%C There are no odd numbers in this sequence.

%C The values for a, b and c are very large. The smallest known solutions contain 81 digits (for k=4).

%C 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

%H Alon Amit, <a href="https://www.quora.com/How-do-you-find-the-positive-integer-solutions-to-frac-x-y%2Bz-%2B-frac-y-z%2Bx-%2B-frac-z-x%2By-4/answer/Alon-Amit">How do you find the positive integer solutions to ...?</a>, Quora, Aug 07, 2017 [Broken link]

%H Andrew Bremner and Allan Macleod, <a href="http://ami.ektf.hu/uploads/papers/finalpdf/AMI_43_from29to41.pdf">An Unusual Cubic Representation Problem</a>, Annales Mathematicae et Informaticae, volume 43 (2014), pages 29-41, see Table 2 page 38.

%H Mathoverflow, <a href="http://mathoverflow.net/questions/227713/estimating-the-size-of-solutions-of-a-diophantine-equation">Estimating the size of solutions of a diophantine equation</a>

%H H. Nakao, <a href="http://www.kaynet.or.jp/~kay/misc/de95.html">Rational Points on Elliptic Curves: x/(y+z)+y/(z+x)+z/(x+y)=n</a>, 2018 (in Japanese).

%H Physics Forums, <a href="https://www.physicsforums.com/threads/find-positive-integer-solutions-to-a-b-c-b-a-c-c-a-b-4.922057/">Find positive integer solutions to a/(b+c)+b/(a+c)+c/(a+b)=4</a>, Aug 06 2017

%H Jinyuan Wang, <a href="/A283564/a283564.txt">PARI program and details of k = 4, 6, 10, 12, 14</a>

%o (Magma)

%o is_A283564 := function(k)

%o     E := EllipticCurve([0, 4*k^2 + 12*k - 3, 0, 32*(k+3), 0]);

%o     return ((Rank(E) eq 1) and (Min([g[1] : g in Generators(E)]) lt 0));

%o end function;

%o [k : k in [1..200] | is_A283564(k)];  // _Robin Visser_, Feb 04 2024

%o (Sage)

%o def is_A283564(k):

%o     E = EllipticCurve([0, 4*k^2 + 12*k - 3, 0, 32*(k+3), 0])

%o     return ((E.rank()==1) and (min([g.xy()[0] for g in E.gens()]) < 0))

%o print([k for k in range(1, 70) if is_A283564(k)])  # _Robin Visser_, Feb 04 2024

%Y Cf. A369896.

%K nonn

%O 1,1

%A _Dmitry Kamenetsky_, Mar 11 2017

%E Definition clarified by _Jimmy Gustafsson_, May 08 2019

%E More terms from _Robin Visser_, Feb 04 2024