Strengthening a Lemma for Proving the Irrationality of the nth Root of 2

[shmible]

I am revising a paper for my proof class. The proof (by contradiction) was: For every natural number n such that n ≥ 2, the n^th root of 2 is irrational.

My lemma was: If a is an even integer, then a² is an even integer.

The feedback for this lemma was that it was wrong, and it was also weak. I was wondering what a stronger lemma, more correct would be.

Perhaps: If a is an even integer, then aⁿ is an even integer.

Is there an even stronger, more correct lemma that can be used to prove my proposition?

Thanks in advance!

 

[micromass]

Why did they say the lemma was wrong?? The lemma is obviously correct in my opinion...

An even stronger result is the following:

If a is an even integer, then aⁿ is divisible by 2ⁿ. But you probably won't need this much strength to prove your theorem...

 

[shmible]

Why did they say the lemma was wrong?? The lemma is obviously correct in my opinion...

An even stronger result is the following:

If a is an even integer, then aⁿ is divisible by 2ⁿ. But you probably won't need this much strength to prove your theorem...

I think it was considered "wrong" because I was using "if a is even, then a² is even" for a more general case of aⁿ.

So do you guys think the lemma: "If a is an even integer, then aⁿ is a even integer" will be fine to use?

What if I made the lemma: "aⁿ is an even integer if and only if a is an even integer" ? This way I can use it in either direction.

EDIT: Perhaps an EVEN MORE general lemma to use would be: "Even integers are closed under multiplication."
I feel that this one is the best so far because it is very broad

 

[chiro]

I think it was considered "wrong" because I was using "if a is even, then a² is even" for a more general case of aⁿ.

So do you guys think the lemma: "If a is an even integer, then aⁿ is a even integer" will be fine to use?

What if I made the lemma: "aⁿ is an even integer if and only if a is an even integer" ? This way I can use it in either direction.

EDIT: Perhaps an EVEN MORE general lemma to use would be: "Even integers are closed under multiplication."
I feel that this one is the best so far because it is very broad

Micromass' solution is probably the best for generalization of the rule. All you have to do is use the fact that a = 2b where b is an integer. So using a^n = (2b)^n = 2^n x b^n, Micromass' suggestion is proved.

 

[Dick]

Uh, the best lemma is the one you would like to use to prove the theorem. What's your strategy? I'm asking because you really don't seem to have one. I would suggest 'if a^n is even then a is even'. Can you prove that?