4 digit Automorphic Numbers

We define a positive integer \(N\) to be an \(n\)-digit automorphic number if the last \(n\) digits of \(N,N^2, N^3,\ldots\) are all equal, and that \(N\) has exactly \(n\) digits.

For example, 376 is a 3-digit automorphic number because \(376^2 = 141\underline{376} , 376^3 = 53157\underline{376}, 376^4=19987173\underline{376} \) and so on.

What is the value of a 4-digit automorphic number?


Inspirations: Here and here.

Automorphic numbers If you want to see a list of automorphic numbers, click me!

On the list, a(n) are the automorphic numbers and n is supposed to represent 1st 2nd 3rd etc.

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