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Suppose I have a sequence:

$$n = \prod_{i=1}^{r(n)} p_i^{d_i}$$

for some primes $p_1 < p_2 < \dots < p_{r(n)}$, and each $d_i \geq 1$ an integer. The function $r(n)$ denotes the number of distinct primes divisor of $n$.

I'm trying to show $r(n) = O(\log n)$.

What I have tried: I think I can establish an upper bound on $\prod_{i=1}^{r(n)} p_i^{d_i}$ to show its asymptotic bound.

To simplify and without loss of generality, let $d_i = 1$. Then,

$$n = \prod_{i=1}^{r(n)} p_i$$ $$\log n = \log (\prod_{i=1}^{r(n)} p_i)$$ $$\log n \geq \Sigma_{i=1}^{r(n)} \log p_i$$

Then not sure where to go from here.

I think I want to show something like:

$$\log n \geq \frac{r(n) + 1}{2}$$

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    $\begingroup$ This seems to be a question about pure Mathematics. What's the computational aspect that you're looking for help with? $\endgroup$ – David Richerby Apr 4 '18 at 15:34
  • $\begingroup$ I wasn't sure if I should post to computer science or to mathematics. I was inclined to post here since it deals with asymptotic bounds. Should I move it? $\endgroup$ – billz Apr 4 '18 at 15:41
  • $\begingroup$ I think so, but you might want to wait to see if anyone else has an opinion. Aaaaand then you got an answer while I was writing my comment, so I guess you may as well leave it. :-) $\endgroup$ – David Richerby Apr 4 '18 at 15:47
  • $\begingroup$ Agree with Dave. This should be migrated to Mathematics. $\endgroup$ – Yuval Filmus Apr 4 '18 at 15:49
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Since each of the prime numbers is at least 2, we have $$n \geq \prod_{i=1}^{r(n)} p_i \geq \prod_{i=1}^{r(n)} 2 = 2^{r(n)}, $$ from which it follows that $r(n) \leq \log_2 n$.

In fact, since the prime numbers are generally larger than 2, we can obtain a better bound, namely $$ r(n) \lesssim \frac{\log n}{\log \log n}. $$ See for example MathWorld, which states the number of distinct prime factors in a primorial.

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  • $\begingroup$ Thank you. I understand your answer of $r(n) \leq \log_2 n$ which is helpful. Could you expand a little bit on the second part, obtaining a better bound? I checked MathWorld (which I did stumble across before) $\endgroup$ – billz Apr 4 '18 at 15:54
  • $\begingroup$ I understand now, just had to reread the MathWorld link...thanks again. $\endgroup$ – billz Apr 4 '18 at 15:58
  • $\begingroup$ We can improve the bound given in the first part using $n \geq q_1 q_2 \dots q_{r(n)}$, where $q_1,q_2,\ldots$ is the sequence of primes. The term on the right-hand side is a primorial. It remains to estimate the rate of growth of primorial. MathWorld states that primorials satisfy $r(n) \sim \frac{\log n}{\log \log n}$. $\endgroup$ – Yuval Filmus Apr 4 '18 at 15:58
  • $\begingroup$ @billz: Instead of answering "what is the maximum number of distinct prime factors of n", you just reverse the question "which is the smallest integer with k or more distinct prime factors". $\endgroup$ – gnasher729 Apr 4 '18 at 19:19

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