# Understand the problem

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# Start with hints

Let

and where .

We have

And since we must have and .

Now, we have to find the solutions from these equations.

The idea is that due to the prime factorization theorem, we can specify that \(x^a = y^b\) leads to a special form of the x and y.

Claim: If for some naturals , then there exists a natural such that and where and .

Consider the prime factorization theorem of the x and y in \(x^a = y^b\). Observe that it implies x and y must have same set of primes by the prime factorization theorem.

Let x and y contain the primes \(p_1, p_2, …, p_k\). Let \( x = \prod_{i=1}^{k} x_i\) and \( y = \prod_{i=1}^{k} y_i\).

The above equation implies that \( a.x_i = b.y_i \). This implies that \( y_i = n.c \) and \( x_i = m.c \), where c is a natural number. Hence \( x = z^m, y = z^n\).

So , we must have wich means and from we have

So, the solutions come out to be and , where is any positive integer.

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