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

In summary, the conversation discusses the possibility of finding solutions for the equation a/(b+c)+b/(a+c)+c/(a+b)=178, with varying values for a, b, and c. It is noted that brute force methods and the use of computers are not effective in finding the smallest solution, which has 80 digits. The conversation also mentions a paper on an unusual cubic representation problem with a discussion on the complications of finding solutions for seemingly simple equations.
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mfb
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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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  • #2
mfb said:
A great example how simple looking problems can have very complicated solutions.
I thought Andrew had already demonstrated it.
 

What does the equation a/(b+c)+b/(a+c)+c/(a+b)=4 mean?

This equation is a mathematical expression that represents the sum of three fractions, with the restriction that the fractions' denominators must be the sum of all three variables. The equation's solution represents a set of positive integer values that can make the equation true.

How do you find positive integer solutions to this equation?

One way to find positive integer solutions to this equation is to use a systematic trial and error approach, where you test different combinations of positive integers for a, b, and c until you find values that satisfy the equation. Another approach is to use algebraic manipulation and number theory concepts to simplify the equation and find patterns in the solutions.

Are there any specific rules or restrictions for the values of a, b, and c in this equation?

Yes, the values of a, b, and c must be positive integers. Additionally, the denominators of the fractions (b+c, a+c, and a+b) must all be positive and non-zero, as division by zero is undefined.

Can this equation have more than one set of positive integer solutions?

Yes, it is possible for this equation to have multiple sets of positive integer solutions. This is because there are often multiple combinations of positive integers that can satisfy the equation. For example, if a=2, b=3, and c=4, the equation would still be true.

Is there a general formula or method for finding all possible positive integer solutions to this equation?

There is no known general formula or method for finding all possible positive integer solutions to this equation. However, number theory concepts such as divisibility, prime factorization, and modular arithmetic can be used to narrow down the possible solutions and make the search for solutions more efficient.

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