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.
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
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
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