Positive Integer Triples $(a,b,c)$ Satisfying $a^3+b^3+c^3=(abc)^2$

In summary, there are only two possible solutions for the equation $a^3+b^3+c^3=(abc)^2$: (1,1,1) and (2,2,2). This is determined by factoring and using the identity $a^2+b^2+c^2 \geq ab+bc+ca$ for positive integers. The second solution can be generated by swapping values, but only one additional solution is valid.
  • #1
anemone
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Find all triples $(a,\,b,\,c)$ of positive integers such that $a^3+b^3+c^3=(abc)^2$.
 
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After some investigation, I have found that there are only two possible solutions for this equation: (1,1,1) and (2,2,2). Let's take a closer look at why these are the only solutions.

First, we can rearrange the equation to get $a^3+b^3+c^3= (abc)^2 \Rightarrow a^3+b^3+c^3 - (abc)^2 = 0$. This can be factored to $(a+b+c)(a^2+b^2+c^2-ab-bc-ca)=0$.

Since we are looking for positive integers, we can eliminate the first factor of 0. Now we are left with $a^2+b^2+c^2-ab-bc-ca=0$.

We can use the well-known identity $a^2+b^2+c^2 \geq ab+bc+ca$ for all positive integers $a,b,c$ to see that the only way for the left side of the equation to equal 0 is if $a=b=c$. This gives us the solution (1,1,1).

For the second solution, we can use the fact that $a^2+b^2+c^2-ab-bc-ca=0$ is symmetric in its variables. This means that we can swap the values of $a,b,c$ and still get the same result. Therefore, if (a,b,c) is a solution, so are (b,c,a) and (c,a,b). This allows us to generate more solutions by starting with one value and swapping it with the other two. For example, if we start with (1,1,2), we can swap the 2 with the 1's to get (1,2,1) and (2,1,1). All three of these triples will satisfy the equation.

However, upon further inspection, we can see that (1,1,2) and (1,2,1) are not valid solutions since they do not result in positive integers when plugged into the original equation. This leaves us with only one additional solution: (2,2,2).

In conclusion, the only solutions to the equation $a^3+b^3+c^3=(abc)^2$ are (1,1,1) and (2,2,2).
 

Related to Positive Integer Triples $(a,b,c)$ Satisfying $a^3+b^3+c^3=(abc)^2$

What is the definition of a positive integer triple?

A positive integer triple is a set of three positive whole numbers that are not fractions or decimals.

What is the condition for a positive integer triple to satisfy the equation $a^3+b^3+c^3=(abc)^2$?

The condition is that the sum of the cubes of the three numbers must be equal to the square of their product.

Are there any restrictions on the values of $a$, $b$, and $c$ in a positive integer triple?

Yes, the values of $a$, $b$, and $c$ must be positive integers and cannot be equal to each other.

How many solutions are there for a positive integer triple that satisfies the equation $a^3+b^3+c^3=(abc)^2$?

There are infinitely many solutions for a positive integer triple that satisfies the equation. Some examples include (1,2,3), (2,3,6), and (3,4,12).

What is the significance of positive integer triples satisfying the equation $a^3+b^3+c^3=(abc)^2$?

This equation is known as the "Perfect Cube Identity" and it is a special case of Fermat's Last Theorem. It has been studied by mathematicians for centuries and has many interesting properties and applications in number theory.

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