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Let us consider the following equation:

Prove that

  • x = log(Pi) = 1.14472988584... is a very good approximation of a solution, up to 10 digits.
  • Using high performance computing or other means, prove that it is correct up to 1,000 digits.
  • Is x = log(Pi) an exact solution?

If the answer to the last question is positive, this would mean that log(Pi) is NOT a transcendental number, but rather, an algebraic number. A remarkable result in itself!

Source for picture: algebraic numbers

Solution and related problem

Any real number larger or equal to 1 is a solution, so there is nothing particular with log(Pi). A more subtle version of this problem is to ask the student to solve the following equation:

We know from the previous problem that if x^5 - x^2 - 1 = x^2 - 1, the equality holds. Thus to find a solution, we just need to solve x^5 - x^2 - 1 = x^2 - 1. The cubic root of 2 is a solution.

More generally, let's define

Then the (unique) real-valued solution to the equation f(x) = 0 is given by

In particular, if p = 3, then x = 2. If p = 2 + log(2) / log(3), then x = 3. Note that the function f is monotonic, and thus invertible. What is the inverse of f?

For related articles from the same author, click here or visit www.VincentGranville.com. Follow me on on LinkedIn, or visit my old web page here.

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Comment by Vincent Granville on September 1, 2018 at 8:31am

A reader posted the following comment:

I think this is a really neat example, a great problem to give to high school students. To solve it, notice that the first two square roots look like sqrt(x + z) - sqrt(x - z). Call this y. Simple algebra gives y = +-sqrt(2x +- 2). Plug in a value, say x = 2, and the answer must be sqrt(2x - 2) as claimed.

Going deeper, let's turn this into a polynomial system. Let z = sqrt(x2 - 1), w = sqrt(2x - 2), v = sqrt(x - z), u = sqrt(x + z). Then we have a system of five equations

(Set each to 0). Use the Dixon resultant to eliminate u, v, w, z. The answer is that x = 1; that is, x must be 1. But this is false! Why? Because the solution is not 0-dimensional. Dixon doesn't have to work then. Nice example.

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