Linked Questions

0
votes
2answers
564 views

Prove that there do not exist nonzero integers $a$ and $b$ such that $a^2=3b^2$. [duplicate]

Well, by intuition, of course there is doesn't exist any nonzero integers, but how would you prove that? I was thinking of doing the GCD of $a$ and $b$ is $1$, but that leads me to nowhere.
0
votes
1answer
346 views

How to prove $\sqrt3$ is irrational? [duplicate]

How to prove $\sqrt3$ is irrational using Fermat's infinite descent method? Like says in Carl Benjamim Boyer's book. Isnt the same prove to $\sqrt2$, in Boyer's book says something like this. $\...
1
vote
1answer
94 views

Positive rationals satisfying: $a^2+a^2=c^2$? [duplicate]

If there are none why not? Thanks in advance.
0
votes
1answer
325 views

Can a^2 = 2b^2 have a solution where a, b are in Z but not zero? [duplicate]

Possible Duplicate: How can you prove that the square root of two is irrational? Can $a^2 = 2b^2$ have a solution where $a, b$ are in $\mathbb{Z}$ but not zero? $\mathbb{Z}$ = positive and ...
-1
votes
2answers
165 views

Direct Irrationality Proof for $\sqrt{3}$ and $\sqrt{6}$ [duplicate]

I am having trouble with proving this directly. I am currently learning about greatest common divisors and know that this has a role in the proof. However, I can only prove the two through ...
128
votes
34answers
19k views

Examples of mathematical results discovered “late”

What are examples of mathematical results that were discovered surprisingly late in history? Maybe the result is a straightforward corollary of an established theorem, or maybe it's just so simple ...
139
votes
12answers
32k views

How to prove: if $a,b \in \mathbb N$, then $a^{1/b}$ is an integer or an irrational number?

It is well known that $\sqrt{2}$ is irrational, and by modifying the proof (replacing 'even' with 'divisible by $3$'), one can prove that $\sqrt{3}$ is irrational, as well. On the other hand, clearly ...
84
votes
18answers
6k views

What is the most unusual proof you know that $\sqrt{2}$ is irrational?

What is the most unusual proof you know that $\sqrt{2}$ is irrational? Here is my favorite: Theorem: $\sqrt{2}$ is irrational. Proof: $3^2-2\cdot 2^2 = 1$. (That's it) That is a ...
27
votes
8answers
34k views

If $a^2$ divides $b^2$, then $a$ divides $b$

Let $a$ and $b$ be positive integers. Prove that: If $a^2$ divides $b^2$, then $a$ divides $b$. Context: the lecturer wrote this up in my notes without proving it, but I can't seem to figure out why ...
11
votes
4answers
3k views

Proof by contradiction: $r - \frac{1}{r} =5\Longrightarrow r$ is irrational?

Prove that any positive real number $r$ satisfying: $r - \frac{1}{r} = 5$ must be irrational. Using the contradiction that the equation must be rational, we set $r= a/b$, where a,b are positive ...
2
votes
7answers
7k views

How to prove that $\sqrt 3$ is an irrational number? [duplicate]

Possible Duplicate: $a^{1/2}$ is either an integer or an irrational number I know how to prove $\sqrt 2$ is an irrational number. Who can tell me that why $\sqrt 3$ is a an irrational number?
6
votes
6answers
4k views

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 ...
2
votes
5answers
296 views

Is $\frac{\sqrt7}{\sqrt[3]15}$ rational or irrational?

Is $\frac{\sqrt7}{\sqrt[3]15}$ rational or irrational? Prove it. I am having a hard time with this question. So far what I did was say, assume it's rational, then $$\frac{\sqrt7}{\sqrt[3]15}=\frac{x}{...
0
votes
2answers
1k views

How to divide it: $\frac{cx+d}{ax+b}$?

Somebody divided $\frac{cx+d}{ax+b}$ into $$ \frac{c}{a} + \frac{d- \frac{bc}{a}}{ax+b} .$$ For use for integrals. Does anybody knows how was it done? Could you show me how to do something like ...
3
votes
1answer
28k views

Is square root of an integer, either an integer or an irrational number, but never (non-integer) rational? [duplicate]

Possible Duplicate: $\sqrt a$ is either an integer or an irrational number. $\sqrt{2}$ is irrational number, but $\sqrt{9} = 3$ is an integer. Are there such integers whose square root is a (non-...

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