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I have trouble in solving a basic exercise of the book Number Theory by Shafarevich and Borevich. It is exercise 4, chapter 1, page 4 in my edition.

It goes as follows: Using the properties of the Legendre-Symbol, show that the congruence $$(x^2-13)(x^2-17)(x^2-221) \equiv 0 \mod m$$ is solvable for all $m$.

I know how to deal with the problem if $m$ is a prime. However, I don't know what to do if $m$ is a power of a prime. (I also know that solving the problem for all prime powers suffices.)

russoo
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The idea is that if neither of $13$ or $17$ are a quadratic residue $\mod p^n$ then $13 \cdot 17 = 221$ must be a quadratic residue (this follows from the group of units of $\mathbb{Z}/p^n \mathbb{Z}$ being cyclic for odd $p$ and you need to consider the case $p = 2$ separately)

Sofía Marlasca Aparicio
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  • Thanks. I see how it works. However, with your approach, I do not really need the Legendre-Symbol at all. Do you have any idea what the authors with their proposal "use the properties of the Legendre-Symbol" might had in mind? – russoo Jul 08 '21 at 16:23
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    That would be another way of proving that the product of two quadratic non-residues is a quadratic residue – Sofía Marlasca Aparicio Jul 08 '21 at 16:26