# Modular square roots of -1

As we have admitted in the last sentence of the first paragraph of the last post that the arsenal of number theory may some day be of utmost help, we try to show some nice related result in this post. 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. Though still, 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.