Fungrim entry: 633265

\left(\left|a\right| \ne \left|b\right| \;\mathbin{\operatorname{and}}\; \left|a\right| \ne \left|2 d\right|\right) \;\implies\; \left(2 d \left|v\right| < \left|a\right|\right)\; \text{ where } \left(d, u, v\right) = \operatorname{xgcd}\!\left(a, b\right)

Assumptions:

a \in \mathbb{Z} \setminus \left\{0\right\} \;\mathbin{\operatorname{and}}\; b \in \mathbb{Z} \setminus \left\{0\right\}

TeX:

\left(\left|a\right| \ne \left|b\right| \;\mathbin{\operatorname{and}}\; \left|a\right| \ne \left|2 d\right|\right) \;\implies\; \left(2 d \left|v\right| < \left|a\right|\right)\; \text{ where } \left(d, u, v\right) = \operatorname{xgcd}\!\left(a, b\right)

a \in \mathbb{Z} \setminus \left\{0\right\} \;\mathbin{\operatorname{and}}\; b \in \mathbb{Z} \setminus \left\{0\right\}

Definitions:

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

Source code for this entry:

Entry(ID("633265"),
    Formula(Where(Implies(And(NotEqual(Abs(a), Abs(b)), NotEqual(Abs(a), Abs(Mul(2, d)))), Less(Mul(Mul(2, d), Abs(v)), Abs(a))), Equal(Tuple(d, u, v), XGCD(a, b)))),
    Variables(a, b),
    Assumptions(And(Element(a, SetMinus(ZZ, Set(0))), Element(b, SetMinus(ZZ, Set(0))))))

Topics using this entry

Greatest common divisor