316. For what natural value of $a$ will the number $a^{2}+$ $+a+1589$ be a perfect square?
316. We find $a=\frac{-1+\sqrt{4 y^{2}-6355}}{2}$, from here $4 y^{2}-$ $-6355=z^{2} ;(2 y+z)(2 y-z)=5 \cdot 31 \cdot 41$.

Answer. $1588,28,316,43$.