## Task A-1.8. (10 points)

Determine all pairs of prime natural numbers $p$ and $q$ for which there exists an integer $a$ such that $a^{4}=p a^{3}+q$.
## Solution.

If $p$ and $q$ are primes and $a$ is an integer such that $a^{4}=p a^{3}+q$, then $a^{3}(a-p)=q$, and we conclude that $a^{3}$ divides $q$.

In particular, $a$ divides $q$, and since $q$ is a prime number, $a$ can only be $1, -1, q$, or $-q$.

Since $a^{3} \mid q$, and it cannot be $q$ or $-q$, it means that $a=1$ or $a=-1$.

If $a=1$, then $1-p=q$, but such prime numbers do not exist. In this case, there is no solution.

If $a=-1$, then $-(-1-p)=q$, which means $1+p=q$.

Thus, exactly one of the prime numbers $p$ and $q$ is even, i.e., equal to 2, so we easily conclude that $p=2$ and $q=3$.

Note: The pair $p=1, q=2$ is not a solution to the problem, as 1 is not a prime number. If a student claims that $(1,2)$ is a solution, they should lose 1 point if they would otherwise have 7 points or more.

## SCHOOL / CITY COMPETITION IN MATHEMATICS

2nd grade - high school - A variant
March 3, 2012.

IF A STUDENT HAS A DIFFERENT APPROACH TO SOLVING THE PROBLEM, THE COMMITTEE IS OBLIGED TO GRADE AND EVALUATE THAT APPROACH APPROPRIATELY.