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### How can you prove that the square root of two is irrational?

I have read a few proofs that $\sqrt{2}$ is irrational. I have never, however, been able to really grasp what they were talking about. Is there a simplified proof that $\sqrt{2}$ is irrational?
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### What are the theorems in mathematics which can be proved using completely different ideas?

I would like to know about theorems which can give different proofs using completely different techniques. Motivation: When I read from the book Proof from the Book, I saw there were many many ...
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### Confused by proof of the irrationality of root 2: if $p^2$ is divisible by $2$, then so is $p$.

In typical proofs of the irrationality of $\sqrt{2}$, I have seen the following logic: If $p^2$ is divisible by $2$, then $p$ is divisible by $2$. Perhaps I am being over-analytical, but how do we ...
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### Show that $3p^2=q^2$ implies $3|p$ and $3|q$

This is a problem from "Introduction to Mathematics - Algebra and Number Systems" (specifically, exercise set 2 #9), which is one of my math texts. Please note that this isn't homework, but I would ...
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### Show that the rationals are an incomplete metric space without reference to reals

I know that you can create rational sequences that converge to irrationals, but is there a simple way to do this without explicit assumption of the existence of the reals? I'm thinking of something ...
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### Can the irrationality of the square root of 2 be proved by using Dirichlet's theorem on primes in an arithmetic progression?

The title says it all. I intend to answer the question myself, in the affirmative. (I would have left the body blank, but the system requires me to post at least 30 characters.)
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### Is there/can there be a model-theoretic proof of this theorem of arithmetic ?

I read on MO that if an integer $a$ is a square mod $p$ for sufficiently large primes $p$, then $a$ is a square. Now that's a statement that looks awfully like a Lefschetz-principle-type statement; ...
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### Prove $\sqrt[m]{D}$ irrationality simply, without unique prime factorization

I apologize if this has been asked already. I am aware of proofs that $\sqrt[m]{D}$ is either integer or irrational for $m,D\in\mathbb{N}$, all of which that I recall and understand make use of the ...
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### The irrationality of $\sqrt[n]{2}$ from the FLT.

It's common to see the Fermat Last Theorem being used to prove the irrationality of $\sqrt[n]{2}$. In fact, according this post, the said proof appeared in American Mathematical Monthly. On the other ...
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### Alternative way of thinking about irrationality of $\sqrt{2}$

A user on stack exchange suggested to think of the following problem as a good way to distinguish between algebraic mindset and an analysis mindset: a) Prove $\sqrt{2}$ is irrational by expressing it ...
### Alternative Proof to irrationality of $\sqrt{2}$ using linear algebra
I am taking my first Proof course, and have been researching alternative proofs to the irrationality of $\sqrt{2}$. One that particularly interested me could be found on this site as number $10$, by ...
In the process of reviewing a proof of the Ax-Grothendieck theorem using ultraproducts, I came across a fun little fact: given a free ultrafilter $\mathcal U$ on the natural numbers, one necessarily ...