Let's say $p_1, p_2, p_3, \ldots $ is the increasing sorted sequence of all prime numbers. Prove that there exists a constant $n_0 \in \mathbb{N}$, so that for all $n \ge n_0$: $$K(p_n) < \log_2(p_n) − 2.$$ Hint: Prime number theorem: $p_n \in \Theta (n \ln n)$.

I think, I could have a program A, which generates $p_n$ in $n\ln n$. I could just give A the index of my prime number and it would generate it.

Another idea of mine was to prove it by contradiction and assume the opposite is true but I can't really make that work. My biggest problem is that I can't find an upper limit to the representation of prime numbers and start from there. Normally I'd just use the factorization for integers…

Any help is appreciated!

  • $\begingroup$ What do you mean by "generates $p_n$ in $n\ln n$"? Neither the prime number theorem nor Kolmogorov complexity is about running time. $\endgroup$ Feb 2 '17 at 13:46
  • $\begingroup$ I pretty much meant what you're stating in your answer but I couldn't really bring it together because I got confused with the running time… $\endgroup$
    – Seen
    Feb 2 '17 at 13:59

Since we can generate $p_n$ given $n$, $K(p_n) \leq K(n) + O(1) \leq \log_2 n + O(1)$. The prime number theorem implies that $p_n = \Theta(n\ln n)$, and so $\log_2(p_n) = \log_2 n + \log_2\log_2 n \pm O(1)$. You take it from here.

  • $\begingroup$ Sorry to bother but tbh I'm actually not that skilled in the use of logarithms. Shouldn't it be log2(n)+log2(ln(n))? Should I solve for n or just replace log2(n) with log2(pn / log2(n)? $\endgroup$
    – Seen
    Feb 2 '17 at 14:08
  • $\begingroup$ It doesn't really matter, since $\ln n = \ln 2 \cdot \log_2 n$. So both expressions coincide, the difference being swallowed by the $O(1)$ error term. $\endgroup$ Feb 2 '17 at 14:17
  • $\begingroup$ Ah ok. I think I got it now :) thanks for the help $\endgroup$
    – Seen
    Feb 2 '17 at 14:42
  • $\begingroup$ @YuvalFilmus, I may misunderstand what your notation is supposed to mean, but I don't see how the difference between $\log_2 n$ and $\ln n$ can be bounded by a constant term for all $n$. $\endgroup$
    – Pontus
    Feb 3 '17 at 9:37
  • $\begingroup$ @Pontus It's bounded by a constant factor. After you take the logarithm, the difference becomes a constant term. $\endgroup$ Feb 3 '17 at 10:40

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