# No positive ℚ = a s.t a*a*a = 2

In summary, the proof provided in the conversation is valid, with the exception of the first equation which is incorrect due to a formatting error. The proof follows the logical steps of showing that if a*a*a = 2, then both j and k must be even, but since a is in lowest form, this is impossible. Therefore, there is no positive rational number a that satisfies the equation.

## Homework Statement

Prove that there is no positive ℚ = a s.t a*a*a= 2

## The Attempt at a Solution

If a = j/k a is in lowest form then one of j or k is odd.

(j^3/k^3) = 2 = j^3=2k^3 letting k^3 = z,

j^3 = 2z so j is even because an even number squared is even, thus an even number cubed is even.

Let j = 2i

so 8i^3=2k^3 = 2(2i^3) = k^3

so k is even for same reason as above.

Because k and j are both even, there is no positive ℚ = a s.t a*a*a= 2Does my proof work?

Something went wrong with formatting of the formulas, especially in the second step, but the overall idea is good.

## Homework Statement

Prove that there is no positive ℚ = a s.t a*a*a= 2

## The Attempt at a Solution

If a = j/k a is in lowest form then one of j or k is odd.

(j^3/k^3) = 2 = j^3=2k^3 letting k^3 = z,

j^3 = 2z so j is even because an even number squared is even, thus an even number cubed is even.

Let j = 2i

so 8i^3=2k^3 = 2(2i^3) = k^3

so k is even for same reason as above.

Because k and j are both even, there is no positive ℚ = a s.t a*a*a= 2Does my proof work?

Your first equation is horribly wrong, just because your were trying to pack too much in a single equation. What you wrote is, essentially, ##A/B=C=A=BC##, which is wrong except when ##C=1## and ##A = B##. What I hope you meant was "##A/B=C##, hence ##A = BC##". Alternatively, you could have written "##A/B=C \Rightarrow A = BC.##" Please tell me you see the difference.

## 1. What is "No positive ℚ = a s.t a*a*a = 2"?

"No positive ℚ = a s.t a*a*a = 2" is a mathematical statement known as the "cube root of 2 problem". It represents the search for a rational number (a number that can be expressed as a ratio of two integers) that, when multiplied by itself three times, results in 2.

## 2. Why is this problem important?

This problem is important because it highlights the limitations of rational numbers in representing all real numbers. It also has connections to other important mathematical concepts such as irrational numbers and the Pythagorean theorem.

## 3. Has anyone solved this problem?

No, this problem remains unsolved. It is known as one of the oldest open problems in mathematics and has been attempted by many mathematicians throughout history, including the ancient Greeks.

## 4. Are there any known solutions that are close to an exact answer?

Yes, there are known approximations for the cube root of 2, such as 1.2599 and 1.2599. However, these are not exact solutions and do not satisfy the equation a*a*a = 2.

## 5. Is there a significance to finding a solution for this problem?

Yes, finding a solution to this problem would have significant implications in mathematics and other fields. It could potentially lead to a better understanding of irrational numbers and their properties, and could also have applications in fields such as cryptography and computer science.

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