User:Salix alba/sandbox

Let ${\textstyle N}$ be odd

• first item
• second item

and ${\textstyle M}$ even.

${\text{ R J M}}$ ${}^{\wedge }$ ${\tilde {\ }}{\tilde {\phantom {a}}}$

${\boldsymbol {\alpha \beta \gamma \delta \epsilon \zeta \eta \theta }}$

${\boldsymbol {\alpha }}$

$(\nabla \times \mathbf {F} )\cdots {\mathrm {d} }\mathbf {S}$ = $\oint _{\partial S}\mathbf {F}$ \cdot${\mathrm {d} }{\boldsymbol {\ell }}$

$a\operatorname {snap} b$

$\displaystyle \sum _{n\in \mathbb {Z} }$ $\displaystyle \sum _{n\in \mathbb {Z} }$

SVG: $\displaystyle \sum _{n\in \mathbb {Z} }$ MathML: ${\sum }_{n\in \mathbb{\mathbb{Z}}}$

\mathcal{A} ${\mathcal {A}}$ ${\mathcal {A}}$ $\mathcal{𝒜}$

${\mathcal {ABCDEFGHIJKLMNOPQRSTUVWXYZ}}$ ${\mathcal {abcdefghijklmnopqrstuvwxyz}}$

$B^{b}|\psi \rangle$ $B^{b}|\psi \rangle$

$B^{b}|\psi )$

$|\psi \rangle$

$|B^{b}\rangle$

$B^{b}\left|\psi \right\rangle$

$B^{b}\left|\psi \right\rangle$

$B^{b}|\psi \rbrace$

$B^{b}|\psi ]$

$p(a|b)$

$p(a|b)$

$\textstyle \max _{B_{y}^{b}|\psi \rangle }$

${\textstyle \max _{B_{y}^{b}|\psi \rangle }}$

$f:X\to Y$ SVG $f:X\to Y$

$A\backslash B$ SVG $A\backslash B$

$A\backslash B$

$A\angle B\sphericalangle C\measuredangle D$

$A\Box B\square C\blacksquare D\Diamond E\lozenge F\blacklozenge G\bigstar H$

$A\triangle B\triangledown C\blacktriangle D\blacktriangledown E$

$A\forall B\exists C\nexists D\And E$

$A\lnot B\neg C\not \operatorname {R} D\bot E\top F$

$A\S B$

$A\diamondsuit B\heartsuit C\clubsuit D\spadesuit E\Game F\flat G\natural H\sharp I$

$A\diagup B\diagdown C$

PNG $A\operatorname {foo} B$ $A\operatorname {foo} B$ PNG $A\operatorname {foo} B$

$\operatorname {G} (V,g)$ SVG $\operatorname {G} (V,g)$

$\|A\|,D(p\|q)$ SVG $\|A\|,D(p\|q)$

$q(v)=\|v\|^{2}$ SVG $q(v)=\|v\|^{2}$

$q(v)=|v|^{2}$ SVG $q(v)=|v|^{2}$

<math>q(v)=|v|^2</math> MathML/MathJax $q(v)=|v|^{2}$ SVG $q(v)=|v|^{2}$

<math>q(v)=\|v\|^2</math> MathML/MathJax $q(v)=\|v\|^{2}$ SVG $q(v)=\|v\|^{2}$

<math>q(v)=\|v\|_A</math> MathML/MathJax $q(v)=\|v\|_{A}$ SVG $q(v)=\|v\|_{A}$

<math>x^2</math> MathML/MathJax $x^{2}$ SVG $x^{2}$

<math>(v)^2</math> MathML/MathJax $(v)^{2}$ SVG $(v)^{2}$

SVG: ${\frac {x}{x}}\quad {\frac {x}{y}}\quad {\frac {x}{l}}\quad {\frac {x}{2}}$ MathML: $\frac{x}{x}\frac{x}{y}\frac{x}{l}\frac{x}{2}$

SVG: ${\frac {x}{x}}\quad {\frac {y}{x}}\quad {\frac {l}{x}}\quad {\frac {2}{x}}$ MathML: $\frac{x}{x}\frac{y}{x}\frac{l}{x}\frac{2}{x}$

<math>\cancel{y}</math> MathML/MathJax ${\cancel {y}}$ SVG ${\cancel {y}}$

<math>\cancel{x}</math> MathML/MathJax ${\cancel {x}}$ SVG ${\cancel {x}}$

<math>\cancel{xyz}</math> MathML/MathJax ${\cancel {xyz}}$ SVG ${\cancel {xyz}}$

let ${\textstyle N}$ be odd

1. foo
2. bar

and ${\textstyle M}$ even.

Pick a random number ${\textstyle 1<a<N}$.|Compute $K=\gcd(a,N)$, the greatest common divisor of $a$ and $N$.|If $K\neq 1$, then $K$ is a nontrivial factor of $N$, with the other factor being ${\textstyle {\frac {N}{K}}}$ and we are done.|Otherwise, use the quantum subroutine to find the order $r$ of $a$.|If $r$ is odd, then go back to step 1.|Compute $g=\gcd(N,a^{r/2}+1)$. If $g$ is nontrivial, the other factor is ${\textstyle {\frac {N}{g}}}$, and we're done. Otherwise, go back to step 1. }}It has been shown that this will be likely to succeed after a few runs. In practice, a single call to the quantum order-finding subroutine is enough to completely factor $N$ with very high probability of success if one uses a more advanced reduction.