Find The Largest N

Algebra Level 5

Consider the function \[f(x) = \dfrac{(x-1)^{1093^{2014}} - x^{1093^{2014}} + 1}{1093(x-1)}.\] Find the last three digits of the largest positive integer \(N\) such that there exist two polynomials \(a(x)\) and \(b(x)\) both having integer coefficients and an integer \(c\) (not necessarily positive) satisfying \[f(x) = (x-c)^N a(x) + b(x).\]

Details and assumptions

  • You might use the fact that \(1093\) is a prime and \(2^{1092}-1\) is a multiple of \(1093^2\) (such primes are called Wieferich primes).

  • This problem is not original.


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