At the ends of the segment, two ones are written. In the middle between them, their sum - the number 2 - is written. Then, in the middle between each pair of adjacent written numbers, their sum is written again, and so on, 1973 times. How many times will the number 1973 be written?

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Let's write down the lines of numbers that form as a result of each step. We will get some table. Let's try to find out how many times the number $n$ will appear in the $n$-th line of this table (it is clear that in each subsequent line - with a number greater than $n$ - the number $n$ will appear exactly as many times as in the $n$-th: all newly formed numbers will already be greater than $n$).

We will say that a pair of natural numbers $(a, b)$ appears in our table if the numbers $a$ and $b$ stand next to each other in one line, with $b$ to the right of $a$.

Lemma. If natural numbers $a$ and $b$ are coprime, then the pair $(a, b)$ appears in the table exactly once; if $a$ and $b$ have a common divisor (other than 1), then the pair $(a, b)$ will not appear in the table at all.

Proof. Induction on $m=\max \{a, b\}$. The base case $(m=2)$ is obvious.

Inductive step. Suppose $a \leq b=m$ (the case $a > b$ is analogous). The pair $(a, b)$ can appear in the table if and only if the pair

$(a, b-a)$ appears in it. The numbers $(a, b)$ and $(a, b-a)$ are either both coprime or not. For the pair $(a, b-a)$, the statement holds by the inductive hypothesis. Therefore, the statement is also true for the pair $(a, b)$.

Each time $n$ is written as the sum of two adjacent numbers $a$ and $b=n-a$, it appears in the pairs $(a, n)$ and $(n, n-a)$. We have proved that this happens exactly once for each $a$ less than $n$ and coprime with it (then, of course, $n-a$ is also coprime with $n$). Thus, each number $n$ will be written in the segment as many times as there are natural numbers less than $n$ and coprime with it.

The number 1973 is prime, so it will appear 1972 times.

## Answer

1972 times.