I am working on proving that $\sqrt{5}$ is irrational. I think I have the proof down, there is just one part I am stuck on. How do I prove that $x^2$ is divisible by 5 then $x$ is also divisible by $5$? Right now I have $5y^2 = x^2$ I am doing a proof by […]

Show that for all $n\in \mathbb{N}$ the number $(\sqrt{2}-1)^n$ is irrational. I do not get the idea of the proof at all, any help appreaciated. edit: I am also thinking whether it will be possible to show $(\sqrt{2}-1)^n=\sqrt{m+1}-\sqrt{m}$

This question already has an answer here: Is $n^{th}$ root of $2$ an irrational number? [duplicate] 2 answers

This is likely overly optimistic, but can one take the results from the Modularity theorem and Ribet’s theorem, and distill these down to an undergrad math level of a way to check if certain rational polynomials have no non-trivial solutions? For instance, I do not understand the details of either theorem, but if I can […]

Let $a,b \in \mathbb{N}^{*}$. Prove that $\sqrt{13a^2+b^2}$ and $\sqrt{a^2+13b^2}$ cannot be simultaneously rational. If $(a,b)=(k,k\cdot6)$, then $\sqrt{13a^2+b^2}$ is rational, but I do not know if those are the only solutions.

Simple proof that $\pi$ is irrational Consider the Gregory – Leibniz series for $\pi/4$: $$\frac \pi 4 = 1 – \frac 1 3 + \frac 1 5 + \cdots $$ Let $A_n/B_n$ be the irreducible fraction given by partial sum $S_n$ up to the $n$th term $\pm 1/(2n-1)$. It can be shown that largest prime […]

I have some difficulties in understanding the proof of “$\sqrt{2}$is irrational” by contradiction. I am reading it in 10th class(in India) Mathematics book( available online, here ) This is the snapshot of it: The proof starts with assuming that $\sqrt{2}$ can be written as a ratio of two integers and then that this fraction can […]

Here is the constructive proof of $\sqrt 2 \not \in \mathbb Q$ found on this page : Given positive integers $a$ and $b$, because the valuation (i.e., highest power of 2 dividing a number) of $2b^2$ is odd, while the valuation of $a^2$ is even, they must be distinct integers; thus $|2 b^2 – a^2| […]

Find conditions on positive integers $a, b, c$ so that $\sqrt{a}+\sqrt{b}+\sqrt{c}$ is irrational. My solution: if $ab$ is not the square of an integer, then the expression is irrational. I find it interesting that $c$ does not come into this at all. My solution is modeled (i.e., copied with modifications) from dexter04’s solution to Prove […]

How can I prove that $\sqrt{3}+ \sqrt{5}+ \sqrt{7}$ is irrational? I know that $\sqrt{3}, \sqrt{5}$ and $\sqrt{7}$ are all irrational and that $\sqrt{3}+\sqrt{5}$, $\sqrt{3}+\sqrt{7}$, $\sqrt{5}+\sqrt{7}$ are all irrational, too. But how can I prove that $\sqrt{3}+ \sqrt{5}+ \sqrt{7}$ is irrational?

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