<s>[INST] Rank of a pair of coprime integers Let's say two pairs of coprime integers $(a, b)$ and $(c, d)$ are connected if $ac + bd = 1$.
A connected chain, or just a chain, is a sequence of coprime pairs in which every two consecutive pairs are connected.
Clearly, if $(a, b)$ is connected to $(c, d)$, then $(a, b)$ is connected to $(c + nb, d - na)$ for an integer $n$.
Using the property, we may find a connected pair $(x, y)$ such that $max(|x|,|y|) < max(|a|,|b|)$.
Continuing the process we may build a chain that ends on $(0, 1)$ or $(1, 0)$ in absolute values.
Examples (in absolute values):

*

*$(41, 53)$, $(22, 17)$, $(7, 9)$, $(4, 3)$, $(1, 1)$, $(0, 1)$.

*$(41, 53)$, $(31, 24)$, $(7, 9)$, $(4, 3)$, $(1, 1)$, $(0, 1)$.

The described algorithm may give chains of different length:

*

*$(41, 61)$, $(58, 39)$, $(2, 3)$, $(1, 1)$, $(0, 1)$.

*$(41, 61)$, $(3, 2)$, $(1, 1)$, $(0, 1)$.

Let's define the rank of a pair of coprime integers $(a, b)$ as the minimal length within all possible chains connecting $(a, b)$ with $(0, 1)$ or $(1, 0)$ in absolute values.
Let's call a chain with the minimal length a minimal chain for $(a, b)$.
Questions:

*

*Does the rank exist for any pair of coprime or prime integers?

*Is there a maximal rank within all pairs of coprime or prime integers with rank?

*Is there an algorithm of constructing a minimal chain or calculating the rank of a given coprime pair?

 [/INST] Your question (3), begin with your coprime pair $(a,b)$  with $a > b > 0,$   the minimal chain comes from using the Extended Euclidean Algorithm and writing the (simple) continued fraction for $\frac{a}{b},$ then let each convergent $\frac{p}{q}$   be part of the chain, as pair $(p,q)$
$$  \gcd( 53, 41 ) = ???    $$
$$ \frac{ 53 }{ 41 } = 1 +  \frac{ 12 }{ 41 } $$
$$ \frac{ 41 }{ 12 } = 3 +  \frac{ 5 }{ 12 } $$
$$ \frac{ 12 }{ 5 } = 2 +  \frac{ 2 }{ 5 } $$
$$ \frac{ 5 }{ 2 } = 2 +  \frac{ 1 }{ 2 } $$
$$ \frac{ 2 }{ 1 } = 2 +  \frac{ 0 }{ 1 } $$
Simple continued fraction tableau:
$$ 
 \begin{array}{cccccccccccc}
 & & 1 & & 3 & & 2 & & 2 & & 2 & \\ 
  \frac{ 0 }{ 1 }   &   \frac{ 1 }{ 0 }   & &   \frac{ 1 }{ 1 }   & &   \frac{ 4 }{ 3 }   & &   \frac{ 9 }{ 7 }   & &   \frac{ 22 }{ 17 }   & &   \frac{ 53 }{ 41 }  
 \end{array}
 $$
$$  $$
$$ 53 \cdot 17 - 41 \cdot 22 = -1 $$
$$  \bigcirc \bigcirc  \bigcirc  \bigcirc \bigcirc  \bigcirc \bigcirc  \bigcirc \bigcirc  \bigcirc     \bigcirc \bigcirc  \bigcirc \bigcirc  \bigcirc \bigcirc  \bigcirc  \bigcirc \bigcirc  \bigcirc \bigcirc  \bigcirc \bigcirc  \bigcirc $$
$$  \gcd( 61, 41 ) = ???    $$
$$ \frac{ 61 }{ 41 } = 1 +  \frac{ 20 }{ 41 } $$
$$ \frac{ 41 }{ 20 } = 2 +  \frac{ 1 }{ 20 } $$
$$ \frac{ 20 }{ 1 } = 20 +  \frac{ 0 }{ 1 } $$
Simple continued fraction tableau:
$$ 
 \begin{array}{cccccccc}
 & & 1 & & 2 & & 20 & \\ 
  \frac{ 0 }{ 1 }   &   \frac{ 1 }{ 0 }   & &   \frac{ 1 }{ 1 }   & &   \frac{ 3 }{ 2 }   & &   \frac{ 61 }{ 41 }  
 \end{array}
 $$
$$  $$
$$ 61 \cdot 2 - 41 \cdot 3 = -1 $$
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