Fungrim entry: 1b47db

\operatorname{xgcd}\!\left(-1, b\right) = \left(1, -\left|\operatorname{sgn}\!\left(\left(b - 1\right) \left(b + 1\right)\right)\right|, \operatorname{sgn}(b) \left(\operatorname{sgn}\!\left(b + 1\right) - \operatorname{sgn}\!\left(b - 1\right)\right)\right)

Assumptions:

b \in \mathbb{Z}

TeX:

\operatorname{xgcd}\!\left(-1, b\right) = \left(1, -\left|\operatorname{sgn}\!\left(\left(b - 1\right) \left(b + 1\right)\right)\right|, \operatorname{sgn}(b) \left(\operatorname{sgn}\!\left(b + 1\right) - \operatorname{sgn}\!\left(b - 1\right)\right)\right)

b \in \mathbb{Z}

Definitions:

Fungrim symbol	Notation	Short description
`XGCD`	$\operatorname{xgcd}\!\left(a, b\right)$	Extended greatest common divisor
`Abs`	$\left\|z\right\|$	Absolute value
`Sign`	$\operatorname{sgn}(z)$	Sign function
`ZZ`	$\mathbb{Z}$	Integers

Source code for this entry:

Entry(ID("1b47db"),
    Formula(Equal(XGCD(-1, b), Tuple(1, Neg(Abs(Sign(Mul(Sub(b, 1), Add(b, 1))))), Mul(Sign(b), Sub(Sign(Add(b, 1)), Sign(Sub(b, 1))))))),
    Variables(b),
    Assumptions(Element(b, ZZ)))

Topics using this entry

Greatest common divisor