## Task 2 - 311212

Determine all triples $(a, b, n)$ of positive integers $a, b, n$ for which the following statements (1) and (2) hold:

(1) The numbers $a$ and $b$ are prime numbers.

(2) It holds that $97 a b = (a+n)(b+n)$.
I. If $a, b, n$ are positive integers that satisfy (1), (2), then:

Due to the uniqueness of the prime factorization and because $a+n>1, b+n>1$ hold, one of the numbers $a+n, b+n$ must be equal to one of the primes 97, $a, b$, and the other must be equal to the product of the two remaining primes. There are only the following cases:

1. $a+n=97 a, b+n=b$,
2. $a+n=97 b, b+n=a$,
3. $a+n=a b, b+n=97$

as well as the three cases obtained by swapping $a, b$.

Case 1 leads to the contradiction $n=0$ and is therefore excluded.

Case 2 leads to $(b+n)+n=97 b, 2 n=96 b, n=48 b, a=b+n=49 b$ and thus to a contradiction to the primality of $a$. Therefore, this case is also excluded.

Case 3 leads to $a+n=a(97-n), n(a+1)=96 a$

Since $a$ is coprime to $a+1$, $n$ must be divisible by $a$, say $n=k \cdot a \quad$ (4) with a positive integer $k$. From (3) it follows that $k(a+1)=96$.

Now 96 has exactly the positive integer divisors $1,2,3,4,6,8,12,16,24,32,48,96$. Of these, only

$$
3, \quad 4 \quad 6, \quad 8, \quad 12, \quad 24, \quad 32, \quad 48
$$

have the form $a+1$ with a prime $a$, namely respectively with

$$
a=2, \quad 3, \quad 5, \quad 7, \quad 11, \quad 23, \quad 31, \quad 47
$$

With this, (5), (4), and the condition $b+n=97$ each lead to

$$
\begin{array}{lllllllll}
k=32, & 24, & 16, & 12, & 8, & 4, & 3, & 2 \\
n=64, & 72, & 80, & 84, & 88, & 92, & 93, & 94 \\
b=33, & 25, & \mathbf{1 7}, & \mathbf{1 3}, & 9, & \mathbf{5}, & , 4, & \mathbf{3}
\end{array}
$$

Only for the highlighted values is $b$ a prime number. Therefore, only the triples

$$
(5,17,80),(7,13,84),(23,5,92),(47,3,94)
$$

and the triples obtained by swapping $a, b$

$$
(17,5,80),(13,7,84),(5,23,92),(3,47,94)
$$

can satisfy conditions (1), (2).

II. They satisfy these conditions, as $3,5,7,13,17,23,47$ are prime numbers and equation (2) is satisfied for each triple.

From I. and II. exactly the triples in (6) and (7) are all the triples of positive integers that satisfy (1) and (2).

Adapted from $[5]$