# Modular square roots of -1

Modular arithmetics plays a central role in mathematics. We may have been very curious the first time we learnt how the modular operators operate. While we are in number theory land, our Sean is working very diligently in mathematical programming land. If we are lucky enough, we should try to merge the two fascinating fields in some future posts.

Most of the time, modular arithmetics theorems are beautiful results on additive, multiplicative modular properties. Aside from that, there are many classical and modern results on the modular square root, the number of them is far less than additive, multiplicative ones.

In other fields such as analysis, square roots of -1 are also pearls of mathematics achievements in history. By asking the same question, namely whether there exists a square root of -1 modulo a natural number $n$, we come up with yet another hard problem, this time of number-theoretic flavor, as shown in this writeup.