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Show that a number is always a perfect square

Source: Argentina Cono Sur TST 2013, Problem 4

January 3, 2015
linear algebramatrixnumber theory proposednumber theory

Problem Statement

Show that the number N=44…4⏟n88…8⏟n−133…3⏟n−12\begin{matrix} \\ N= \end{matrix} \underbrace{44 \ldots 4}_{n} \underbrace{88 \ldots 8}_{n} - 1\underbrace{33 \ldots3 }_{n-1}2 is a perfect square for all positive integers nn.
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