Exercise 8. Determine the sum of all integers $a$ such that $a^{2}-82 a$ is a prime number.
The solution to exercise 8 is $a(a-82)$ is prime, so one of the factors is equal to 1 or -1. If $a=1$ then $a(a-82)$ is negative, so it is not prime. If $a=-1$ then $a(a-82)=83$ is prime. If $a-82=1$ then $a=83$ so $a(a-82)$ is prime. Finally, if $a-82=-1$ then $a(a-82)<0$ is not prime. The answer is therefore $-1+83=82$.