31 Oct '07 13:212 edits

n is a positive integer. Here is a proof that sqrt(n) can't be a non-integer rational number:

Assume this is not the case. Then there is a minimal positive integer k such that k*sqrt(n) is an integer. But k*sqrt(n)-k*floor(sqrt(n)) is a smaller positive integer having the same property.

My question: can you write a SIMILAR proof, that will show that for all positive integers m and n, the mth root of n can't be a non-integer rational? It is easy to prove this theorem if we are allowed to use the fundamental theorem of arithmetic (=every integer has a unique prime factorization). But I want a proof similar to the proof I gave for square roots.

Thanx ðŸ™‚

Assume this is not the case. Then there is a minimal positive integer k such that k*sqrt(n) is an integer. But k*sqrt(n)-k*floor(sqrt(n)) is a smaller positive integer having the same property.

My question: can you write a SIMILAR proof, that will show that for all positive integers m and n, the mth root of n can't be a non-integer rational? It is easy to prove this theorem if we are allowed to use the fundamental theorem of arithmetic (=every integer has a unique prime factorization). But I want a proof similar to the proof I gave for square roots.

Thanx ðŸ™‚