# Finding liminf of p_n/n where p_n = nth prime

• Dragonfall
In summary, the question is asking to find the lim inf of p_n/n where p_n is the nth prime. It is known that p_n ~ n logn and the question is whether a simple substitution would work. It is also discussed if the sequence is monotone decreasing for large n. The answer is that the substitution is justified and the first question is true. The second question is unknown. An equivalent question is whether np_{n+1}-(n+1)p_n<0 for only finitely many n.
Dragonfall

## Homework Statement

Find the lim inf of p_n/n where p_n is the nth prime.

## Homework Equations

Well p_n ~ n logn, but I'm not sure if a simple substitution would work. This question may be incredibly trivial or open, and I can't figure out which.

I'm also wondering if the sequence above is monotone decreasing for sufficiently large n (this is not true for small n).

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You have to think what $$P_n$$~ $$n \log n$$ actually means!

It means the quotient of the 2 functions as n approaches infinity is 1, note this does not mean the difference of the functions as n approaches infinity is 0.

eg n+2~n, but the difference of the functions will always be 2.

Anywho, so we know $$\lim_{n\rightarrow {\infty}} \frac{P_n}{n\log n} = 1$$.

Since we want $$\lim_{n\rightarrow {\infty}} \frac{P_n}{n}$$, we can make the simple substitution and get log n.

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Ok I wrote that backwards. I want n/p_n, not p_n/n. The question is whether lim n/(nlog n)=0 implies that lim inf n/p_n=0. More importantly is whether n/p_n is monotone for large n.

Well as per my previous post, the substitution is justified. So Your first question is true, it implies it. The 2nd part, I would not know.

An equivalent question is whether $$np_{n+1}-(n+1)p_n<0$$ for only finitely many n.

## 1. What is the definition of liminf?

The liminf, or limit inferior, of a sequence is the smallest limit point of the sequence. In other words, it is the greatest lower bound of the set of all possible subsequential limits.

## 2. What is p_n and why is it important?

p_n is the nth prime number in the sequence of prime numbers. It is important because it helps us understand the distribution and behavior of prime numbers, which has been a topic of interest in mathematics for centuries.

## 3. How is liminf of p_n/n related to the Prime Number Theorem?

The Prime Number Theorem states that the number of primes less than or equal to x is approximately equal to x/ln(x). The liminf of p_n/n is a way to measure how closely the sequence of prime numbers follows this approximation. If the liminf is equal to 1, then the sequence follows the Prime Number Theorem perfectly.

## 4. How can we calculate the liminf of p_n/n?

Unfortunately, there is no known formula for calculating the liminf of p_n/n. It can only be approximated by computing the values of p_n/n for increasingly large values of n.

## 5. What are some applications of knowing the liminf of p_n/n?

Knowing the liminf of p_n/n can have implications in number theory and cryptography. It can also help us better understand the distribution of prime numbers and potentially lead to new insights in this area of mathematics.

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