Question 29
Let $a \gt b \gt 1$ be relatively prime positive integers. Define the weight of an integer $c$, denoted by $w(c)$, to be the minimal possible value of $|x| + |y|$ taken over all pairs of integers $x$ and $y$ such that
$$
a x + b y = c.
$$
An integer $c$ is called a local champion if $w(c) \geq w(c \pm a)$ and $w(c) \geq w(c \pm b)$.
Find all local champions and determine their number.
$$
a x + b y = c.
$$
An integer $c$ is called a local champion if $w(c) \geq w(c \pm a)$ and $w(c) \geq w(c \pm b)$.
Find all local champions and determine their number.