## Task 19/80

Determine all solutions of the equation $p^{2}+576=n^{2}$, where $p$ is any prime number and $n$ is any natural number.
From $p^{2}+576=n^{2}$ it follows that

$$
p^{2}=n^{2}-576=n^{2}-24^{2}=(n-24)(n+24)
$$

From the uniqueness of the prime factorization, it follows that because $n-24 \neq n+24$,

$$
n-24=1 \rightarrow n=25 \quad ; \quad p^{2}=n+24 \rightarrow p^{2}=25+24=49=7^{2}
$$

which indeed yields a prime number with $p=7$. The solution path excludes other possibilities. Therefore, $(p ; n)=(7 ; 25)$ is the only solution pair.