Divisibility #1

A polynomial f(x)f(x) satisfies the following conditions:

  • limxf(x)x2=1\displaystyle \lim_{x \to \infty} \frac{f(x)}{x^2} = 1
  • f(1)>1f(-1) > 1
  • f(0)f(0) is a prime number
  • There exists an integer k>1k > 1 such that for all positive integer nn, f(n)f(n) is a positive integer divisible by kk

Find the maximum value of f(1)+kf(1) + k.


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