What are quadratic residues?

Quadratic residues are like special numbers that come up when you square other numbers and do some simple math.

Imagine you have a bag of marbles, and every time you take out a marble, you count how many squares it can make. Some numbers are super common in this game, those are your quadratic residues.

When you square a number, you're just multiplying it by itself. For example:

• 2 squared is 4
• 3 squared is 9
• 4 squared is 16

These results, like 4, 9, and 16, are the squares of those numbers.

Now imagine you're playing with marbles on a number line. When you square a number, you might not always land exactly where you want, sometimes there's a little extra or missing. That extra or missing part is called a remainder.

Quadratic residues are the remainders that show up most often when you divide these squares by a certain number. It’s like looking at how many marbles spill over after each turn, those spilled-over numbers are your quadratic residues.