True fact: $16080 = 127^2 – 7^2$. Without doing any factoring or computations besides a bit of adding and subtracting, factor $16080$ as a product of two integers. (Hint on discord if you are stuck.)

Demonstrate the p-1 factoring method on $n=5063$. Show the structure of the calculation, but you can use Sage for modular exponentiation, gcd’s, and division. (You can write a little Sage for-loop to do this, but write up the results in such a way that it demonstrates the method.)

Try to demonstrate the p-1 factoring method for $n=5963$. It won’t work… what goes wrong?

Why did the last example fail? Can you design another $n$ for which this method fails?