Curious numbers

A an n-digit number is said to be curious if the last n digits of its square are the same as the original number. For example, 25² = 625 and 76² = 5776. (Curious numbers are also known as automorphic numbers.)

There are bigger curious numbers, such as 212890625 and 787109376:

212890625² = 45322418212890625

and

787109376² = 619541169787109376.

And if the square of x has the same last n digits as x, so does the cube of x and all higher powers.

It turns out that for each n > 1, there are two curious numbers of length n.

Always two there are; no more, no less. - Yoda

There's even a formula for the two solutions. The first is the remainder when 5 to the power 2ⁿ is divided by 10ⁿ and the second is 10ⁿ + 1 minus the first.

Here's a little Python code to show that the first several solutions really are solutions.

for i in range(2, 20): a = 5**(2**i) % 10**i b = 10**i - a + 1 print((a**2 - a)%10**i, (b**2 - b)%10**i)