Intereting Posts

Converting a function for “velocity vs. position”, $v(x)$, to “position vs. time”, $p(t)$
Why are rotational matrices not commutative?
Show that the sequence $\sqrt{2},\sqrt{2\sqrt{2}},\sqrt{2\sqrt{2\sqrt{2}}},…$ converges and find its limit.
Character and representations of $\Bbb C^\times$
Why is the 'change-of-basis matrix' called such?
Integration of $x^2 \sin(x)$ by parts
1-1 correspondence between and
Continuous and bounded imply uniform continuity?
Probability, integers and reals (soft question)
Arrow's Impossibility Theorem and Ultrafilters. References
If $\sqrt{a} + \sqrt{b}$ is rational then prove $\sqrt{a}$ and $\sqrt{b}$ are rational
isometry $f:X\to X$ is onto if $X$ is compact
Neighborhoods vs Open Neighborhoods?
Why is there apparently no general notion of structure-homomorphism?
Given a torsion $R$-module $A$ where $R$ is an integral domain, $\mathrm{Tor}_n^R(A,B)$ is also torsion.

All the units are satisfied Pell’s equation $a^2-2b^2=\pm1$ for $\mathbf{Z}[\sqrt{2}]$, $a,b\in\mathbf{Z}$.

Here is my proof:

Let $a+b\sqrt{2}$ be a unit $\in\mathbf{Z}[\sqrt{2}]$. This implies

$$(a+b\sqrt{2})(c+d\sqrt{2})=1, c+d\sqrt{2}\in\mathbf{Z}\sqrt{2}$$

$$\implies \mathrm{norm}((a+b\sqrt{2})(c+d\sqrt{2}))=1$$

$$\implies \mathrm{norm}(a+b\sqrt{2})\cdot \mathrm{norm}(c+d\sqrt{2})=1$$

$$\implies (a+b\sqrt{2})(a-b\sqrt{2})(c+d\sqrt{2})(c-d\sqrt{2})=1$$

$$\implies (a^2-2b^2)(c^2-2d^2)=1$$

$$\implies \left[ a^2-2b^2=1\text{ and } {c}^2-2{d}^2=1\right]\text{ or } \left[ a^2-2{b}^2=-1\text{ and }{c}^2-2{d}^2=-1\right]$$

$$\implies {a}^2-2{b}^2=\pm1$$

- Integral solutions to $1\times2+2\times3+\cdots+m\times(m+1)=n\times(n+1)$.
- $x^y = y^x$ for integers $x$ and $y$
- Solutions to $ax^2 + by^2 = cz^2$
- Numbers of the form $\frac{xyz}{x+y+z}$
- If $n\in\mathbb N$ and $k\in\mathbb Z$, solve $n^3-32n^2+n=k^2$.
- Odd binomial sum equality has only trivial solution?

- Inverse modulo question?
- How would I find a number where $\sum_{d\mid n}d > 100n$?
- Proof that a Combination is an integer
- Prove or disprove: For every integer a, if a is not congruent to 0 (mod 3), the a^2 is congruent to 1 (mod 3)
- Sum of the digits of a perfect square
- Can this approach to showing no positive integer solutions to $p^n = x^3 + y^3$ be generalized?
- Find remainder of $F_n$ when divided by $5$
- To what divisors $a$ of $n$ can Euler's Theorem multiplied by $a$ be generalized, i.e. when is $a^{\phi(n)+1}\equiv a \pmod n$?
- When is a sum of consecutive squares equal to a square?
- Euclidean Algorithm - find $\gcd(172, 20)$ and solve $172a + 20b = 1000$.

- Cardinality of Cartesian Product of finite sets.
- Infinite subset with pairwise comprime elements
- A simple explanation of eigenvectors and eigenvalues with 'big picture' ideas of why on earth they matter
- If a prime $p\mid ab$, then $p\mid a$ or $p\mid b$
- How does one find a formula for the recurrence relation $a_{1}=1,a_{2}=3, a_{n+2}=a_{n+1}+a_{n}?$
- How do I divide a function into even and odd sections?
- How to integrate $\int\frac{1}{\sqrt{1+x^3}}\mathrm dx$?
- Infinite Series -: $\psi(s)=\psi(0)+\psi_1(0)s+\psi_2(0)\frac{s^2}{2!}+\psi_3(0)\frac{s^3}{3!}+.+.+ $.
- Solving Systems of ODEs
- What is going on in this degree 8 number field that fails to be a quaternion extension of $\mathbb{Q}$?
- $e^{i\theta}$ $=$ $\cos \theta + i \sin \theta$, a definition or theorem?
- Inequality Of Four Variables
- Computing the ring of integers of a number field $K/\mathbb{Q}$: Is $\mathcal{O}_K$ always equal to $\mathbb{Z}$ for some $\alpha$?
- Recast the scalar SPDE $du_t(Φ_t(x))=f_t(Φ_t(x))dt+∇ u_t(Φ_t(x))⋅ξ_t(Φ_t(x))dW_t$ into a SDE in an infinite dimensional function space.
- Motivation behind the definition of localization