Ridiculous Bases!

Number Theory Level pending

What is \((31^2)_n\)?

Where, \(n = \binom{m}{13}\) and \(m = \binom{82}{54}\)

Please provide your answer in base \(n\), not decimal.

Clarification: The subscript means base \(n\), and the notation \(\binom{a}{b}\) indicates "\(a\) choose \(b\)" or the binomial coefficient indexed by \(a\) and \(b\).

Hint: Although this problem appears to be compute intensive, in reality it is not... Good luck!

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