求解非线性方程n/ln(n)=990000（第990000个素数近似式）

Why the Actual Prime Doesn't Satisfy Your Transformed Equation #

First, let's clear up the core issue: the formula n / ln(n) = k (where k is the prime's index) is an approximation, not an exact equality. This comes from the Prime Number Theorem (PNT), which tells us the prime-counting function π(x) (number of primes ≤ x) follows:

π(x) ~ x / ln(x)

The ~ symbol means "asymptotically equivalent"—as x grows very large, the ratio of π(x) to x/ln(x) approaches 1, but they never perfectly match for any finite x. For the k-th prime p_k, the PNT gives the asymptotic estimate p_k ~ k ln(k), which rearranges to k ~ p_k / ln(p_k). Your original equation is just this approximation rearranged, so it's never going to hold exactly for the actual prime value.

Your algebraic manipulation (rearranging to n = e^(n/990000)) is mathematically correct, but since the starting equation is an approximation, the transformed version is also an approximation. That's why substituting the real 990000th prime gives a mismatch—this is normal, not a mistake in your steps.

Correct Methods to Find (or Approximate) the k-th Prime #

1. Use Improved Asymptotic Approximations #

The basic PNT estimate is rough, but we can add higher-order terms for far better accuracy. A common improved formula for the k-th prime is:

p_k ≈ k (ln k + ln ln k)

For k=990000:

• ln(990000) ≈ 13.809
• ln(ln(990000)) ≈ ln(13.809) ≈ 2.626
• Calculating gives p_k ≈ 990000*(13.809 + 2.626) ≈ 16270650

For even closer results, use this higher-order expansion:

p_k ≈ k (ln k + ln ln k - 1 + (ln ln k - 2)/ln k)

This will get you much nearer to the actual prime value 15318907.

2. Numerical Iteration for the Approximation Equation #

If you want to solve the approximate equation n / ln(n) = 990000 precisely (to get the best estimate from that formula), use the Newton-Raphson method—this is the standard way to solve transcendental equations with no closed-form solution.

Step-by-Step Newton-Raphson:

1. Define the function we need to find the root of:

   f(n) = n / ln(n) - 990000
   

2. Compute its derivative:

   f’(n) = (ln(n) - 1) / (ln(n))²
   

3. Start with an initial guess (use the improved approximation above, e.g., n₀ = 16270650)
4. Iterate using this formula:

   n_{i+1} = n_i - f(n_i)/f’(n_i)
   

   After a few iterations, you'll converge to a value around ~15.9 million—still close to the actual prime, but not identical, since the original equation only approximates the prime-counting function.

3. Exact Calculation #

To get the exact k-th prime, you'd need to use a specialized prime-counting algorithm (like Meissel-Lehmer) paired with a search for the smallest n where π(n) = 990000, or reference precomputed prime tables. This is how the exact value 15318907 was determined.

内容的提问来源于stack exchange，提问作者0xI