Graduate Find positive integer solutions to a/(b+c)+b/(a+c)+c/(a+b)=4

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SUMMARY

The equation a/(b+c) + b/(a+c) + c/(a+b) = 4 has been analyzed, revealing that while negative integers can yield solutions, such as a=4, b=-1, c=11, the smallest positive integer solution is extraordinarily large, consisting of 80 digits. Further exploration of the equation with different integers, such as replacing 4 with 178, leads to integer solutions with even larger minimal representations, reaching nearly 400 million digits. This discussion highlights the complexity that can arise from seemingly simple mathematical problems.

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What an innocently looking equation.
If we allow negative integers, a=4, b=-1, c=11 is a solution.

Do some tricks with divisibility?
Solve for a?
Brute force with the computer?It won't help. There are solutions, but the smallest solution has 80 digits.

What happens if we replace 4 by other integers?
a/(b+c)+b/(a+c)+c/(a+b)=178?
There are integer solutions, but the smallest one has nearly 400 million digits.

A great example how simple looking problems can have very complicated solutions.

Paper: An unusual cubic representation problem (PDF)
Here is a discussion
 
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mfb said:
A great example how simple looking problems can have very complicated solutions.
I thought Andrew had already demonstrated it.
 

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