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Advanced factorization is a gateway to algebraic number theory, which mathematicians study in order to solve famous conjectures like Fermat's Last Theorem.

Consider the number

\[ N = \overline{123456789101112\ldots 9899100}, \]

which is obtained from writing all the integers from 1 to 100 and removing the spaces that are in between. What are the first 3 digits of \(N^2 \)?

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Evaluate

\[ \lfloor \left( \sqrt{ 23 } + \sqrt{17 } \right)^4 \rfloor . \]

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Integers \(N\) and \(M\) satisfy \( 0 \leq M \leq 21! \) and

\[ \sum_{i=1}^{21} (i^2 + i + 1) i ! = N \times 21 ! - M. \]

What is \(N + M \)?

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If \( x = \sqrt{ 11 + \sqrt{ 7 } } \) and \( y = \sqrt{ 11 - \sqrt{ 7 } } \), evaluate

\[ x^6 + y^6. \]

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