Articles of prime numbers

Proof that the derivative of the prime counting function is the probability of prime?

The derivative of the estimation of the prime counting function, $\frac{x}{ln(x)}$, is $\frac{ln(x)-1}{ln(x)^2}$, which is approximately $\frac{1}{lnx}$ for large values of $x$. According to the prime number theory, $\frac{1}{lnx}$ is the probability that a randomly chosen integer between 2 and $x$ is a prime number. Why is the derivative of the prime counting function the […]

What is the smallest number k, such that $k^{2014}+2014$ is prime?

What is the smallest number k, such that $$k^{2014}+2014$$ is prime ? I checked upto $k= 24000$ and did not find a prime. Since the numbers do not grow very fast ($k=92204$ produces a $10 000$-digit number), the smallest k should be not too big.

Is this observation equivalent to Legendre's conjecture? $\forall n \in \Bbb N^+, \exists k \in \Bbb N^+:\ \ (n^2)\# \lt k! \lt (n+1)^2\#$

Is this observation equivalent to Legendre’s conjecture? $$E1:\forall k \in \Bbb N^+ \not \exists n,m \in \Bbb N^+, n \not = m:\ \ \ k! \le n^2\# \lt m^2\# \le (k+1)! $$ This observation is a consequence of some tests regarding the kind answer of @Will Jagy to a previous question at MSE. $E1$ can […]

Reverse of Chinese Remainder Theorem

For the following: $(102n-51) \not\equiv 2 \pmod {2,3,5,7,11,13,…,\sqrt{102n-51}}$ (That’s probably completely incorrect use of symbols, but I mean not equivalent to 2 mod any prime less than $\sqrt{102n-51}$) here are my questions: What is the first (smallest) $n$ solution? Are there infinitely many $n$ solutions? (Most importantly) Is there a way we’d know (be able […]

Is this a valid partial refinement of Ingham's upper bound for prime gaps?

This is a follow-up question from this one that was kindly answered by @JordanPayette. The corrections were applied for this solution. Let $p_n$ denote the $n^{th}$ prime number. Ingham showed that: $$p_{n+1} – p_n \lt K p_n^{\frac{5}{8}}$$ where $K$ is a fixed positive integer, is an upper bound for the prime gaps. (A.E.Ingham, On the […]

A composite odd number, not being a power of $3$, is a fermat-pseudoprime to some base

I want to prove the following statement : If $n$ is an odd composite number, not being a power of $3$ (or, equivalent, having a prime factor $p>3$), $n$ is a fermat-pseudoprime to same base $a$, in other words, there is a number $a$ with $$1<a<n-1$$ and $$a^{n-1}\equiv 1\ (\ mod\ n\ )$$ I was […]

Iterated Pi function

Does anyone have any information on iterating the prime counting function. Specifically, $\pi_n(x)$=$\pi(\pi_{n-1}(x))$, and $\pi_1(x)$=$\pi(x)$. I’m looking for anything on this function, what it may be called (when I search for iterate pi function, all I get is information about calculating pi).

How to prove that the partial Euler product of primes less than or equal x is bounded from below by log(x)?

How does one prove $\prod_{p \leq x}(1 – \frac{1}{p})^{-1} \geq \log(x)$?

Prove about prime numbers obtained from certain sums of squares of an integer $n$

I would like to ask for a prove about an observation I did regarding the sums of squares and prime numbers (in another question here), or a counterexample of it. My capabilities to do this kind of demonstrations are very poor, so I do not know if in this case this is difficult to prove […]

How do I find(isolate) the n-th prime number?

So I wanted to solve this SPOJ problem and I did some research about finding the n-th prime number. This formula came across and it stated that the n-th prime must be in this range: $n \ln n + n(\ln\ln n – 1) < p_n < n \ln n + n \ln \ln n$ for […]