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If m,k,n are natural numbers and n>1, prove that we cannot have m(m+1)=k^{n}.

My attempt :

Using induction:

If m follows this rule... m+1 must follow it ..

so

(m+1)(m+2) = k^{n}

Since every natural number can be expressed as the product of primes, it follows that (m+1) and (m+2) are primes.

Now, my question is...

are there any two consecutive primes which can be expressed in the form of k^{n}.

Thanks for any help.

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# A mathematics olympiad problem

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