$\left(\left|a\right| \ne \left|b\right| \;\mathbin{\operatorname{and}}\; \left|a\right| = \left|2 d\right|\right) \;\implies\; \left(v = \operatorname{sgn}(b)\right)\; \text{ where } \left(d, u, v\right) = \operatorname{xgcd}\!\left(a, b\right)$
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| = \left|2 d\right|\right) \;\implies\; \left(v = \operatorname{sgn}(b)\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
Sign$\operatorname{sgn}(z)$ Sign function
XGCD$\operatorname{xgcd}\!\left(a, b\right)$ Extended greatest common divisor
ZZ$\mathbb{Z}$ Integers
Source code for this entry:
Entry(ID("4e5aad"),
Formula(Where(Implies(And(NotEqual(Abs(a), Abs(b)), Equal(Abs(a), Abs(Mul(2, d)))), Equal(v, Sign(b))), 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

Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2021-03-15 19:12:00.328586 UTC