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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.

%I A283564 #60 Jun 18 2025 18:06:47
%S A283564 4,6,10,12,14,16,18,24,28,32,38,42,46,48,58,60,66,76,82,92,102,112,
%T A283564 116,126,130,132,136,146,156,158,162,178,182,184,186,196,198,200,206,
%U A283564 218,232,266,270,276,282,304,310,312,314,318,332,336,338,346,348,362,364,378,382,388,402,408
%N 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.
%C A283564 There are no odd numbers in this sequence.
%C A283564 The values for a, b and c are very large. The smallest known solutions contain 81 digits (for k=4).
%C A283564 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 A283564 Xianwen Wang, <a href="/A283564/b283564.txt">Table of n, a(n) for n = 1..107</a> (for a(n)<=1000).
%H A283564 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 A283564 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 A283564 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 A283564 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 A283564 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 A283564 Jinyuan Wang, <a href="/A283564/a283564.txt">PARI program and details of k = 4, 6, 10, 12, 14</a>
%o A283564 (Magma)
%o A283564 is_A283564 := function(k)
%o A283564     E := EllipticCurve([0, 4*k^2 + 12*k - 3, 0, 32*(k+3), 0]);
%o A283564     return ((Rank(E) eq 1) and (Min([g[1] : g in Generators(E)]) lt 0));
%o A283564 end function;
%o A283564 [k : k in [1..200] | is_A283564(k)];  // _Robin Visser_, Feb 04 2024
%o A283564 (Sage)
%o A283564 def is_A283564(k):
%o A283564     E = EllipticCurve([0, 4*k^2 + 12*k - 3, 0, 32*(k+3), 0])
%o A283564     return ((E.rank()==1) and (min([g.xy()[0] for g in E.gens()]) < 0))
%o A283564 print([k for k in range(1, 70) if is_A283564(k)])  # _Robin Visser_, Feb 04 2024
%Y A283564 Cf. A369896.
%K A283564 nonn
%O A283564 1,1
%A A283564 _Dmitry Kamenetsky_, Mar 11 2017
%E A283564 Definition clarified by _Jimmy Gustafsson_, May 08 2019
%E A283564 More terms from _Robin Visser_, Feb 04 2024