Possible values of $k$ in the given equation

Question

The equation $x^2-58x+k$ has both the roots prime and belong to integers. What are the possible values of $k$?

I tried writing down 58 in all sums of two primes but I feel the list will go on.

Answer 1

Viète's formulas tell us that the sum of roots is 58 and that the product of roots is $k$. All the decompositions of 58 into two primes follow, with the corresponding values of $k$. $$5+53,k=265$$ $$11+47,k=517$$ $$17+41,k=697$$ $$29+29,k=841$$ Thus, these are all the possible values for $k$.

Answer 2

Note that the sum of the roots is $58=5+53=11+47=17+41=29+29$. Hence $k$, the product of the roots, can be $5\cdot 53$, $11\cdot 47$, $17\cdot 41$, $29^2$.