## Task 24/87

For which prime numbers $p$ does the system of equations

$$
\begin{array}{r}
x+\log _{2} y=11 p \\
2 p-\log _{2} p=11-z \\
z+\log _{2} y=x-8
\end{array}
$$

have integer solution triples $(x ; y ; z)$, and what are these?
We assume that the system of equations is solvable in integers and $(x ; y ; z)$ is such a solution triplet. From equation (2), it follows that, since $11-z=g$ is an integer, the equations

$$
2 p-g=\log _{2} p \quad ; \quad p=2^{2 p-g}=g^{\prime}
$$

hold, where $g^{\prime}$ is also an integer. Since $p$ is a prime number, $p=2$ and thus $g=3$ must hold. From equation (2), it immediately follows that $z=8$. Thus, equations (1) and (3) take the form

$$
x+\log _{2} y=22 \quad ; \quad \log _{2} y=x-16
$$

Substituting the second equation into the first yields $x=19$. Consequently,

$$
\log _{2} y=3 \quad ; \quad y=2^{3}=8
$$

Thus, the integer solution triplet of the system of equations is $(19 ; 8 ; 8)$.

### 2.28 Problems and Solutions 1988