The problem of finding the smallest positive. $ a, b, c $ for which $ a / (b + c) + b / (a + c) + c / (a + b) = 4 $ It turns out to be surprisingly difficult, and has done the rounds on the Internet and social networks, and Andrew Bremner and Allan Macleod have written an article about it. By looking at the associated elliptic curve, you can deduce that the smallest solution is begin {align}

a & = 154476802108746166441951315019919837485664325669565431700026634898253202035277999 \

b & = 4373612677928697257861252602371390152816537558161613618621437993378423467772036 \

c & = 36875131794129999827197811565225474825492979968971970996283137471637224634055579

end {align} and permutations. Another interesting question is to find the smallest one. $ a, b, c $ (in a certain sense) for which $ a / (b + c) + b / (a + c) + c / (a + b) $ is a whole number $> 2 $ ($ a = 1, b = 1, c = 3 $ satisfy $ N = 2 $). I think it can be

begin {align}

a & = 6383088000457968550863626020707964592827 \

b & = 86901761472912010754925122912564100123 \

c & = 2743260400516683056616306684496286550899825

end {align}

that produce the value $ N = 424 $. One of the reasons why these numbers are so small is that the associated elliptic curve has the rank 3, therefore, it has a great flexibility when combining the generators to obtain a point in the region of the non-identity component that is assigned A positive $ a, b, c $. I'm not sure of the smallest value of $ N $ associated with an elliptic curve of rank 4, but it can be $ N = 13502 $.