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, 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

Author

Dmitry Kamenetsky, Mar 11 2017

Keywords

• nonn

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

• Xianwen Wang, Table of n, a(n) for n = 1..107 (for a(n)<=1000).
• 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.
• Mathoverflow, Estimating the size of solutions of a diophantine equation
• H. Nakao, Rational Points on Elliptic Curves: x/(y+z)+y/(z+x)+z/(x+y)=n, 2018 (in Japanese).
• Physics Forums, Find positive integer solutions to a/(b+c)+b/(a+c)+c/(a+b)=4, Aug 06 2017
• Jinyuan Wang, PARI program and details of k = 4, 6, 10, 12, 14

Crossrefs

Cf. A369896.

Programs

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

Extensions

Definition clarified by Jimmy Gustafsson, May 08 2019

More terms from Robin Visser, Feb 04 2024