3
votes
0answers
36 views

An Inequality with Prime Numbers

Let $m\geq 8$ be an integer, and let $q$ be the smallest prime that is greater than $m$. Let $p$ be a prime that satisfies $q\leq p<q^2$, and let $p_0$ be the smallest prime such that ...
1
vote
0answers
33 views

Prime bounds under RH

Continuing from here, since $$ \sum_{k=1}^{\lfloor\log(n)\rfloor}\dfrac{\pi(n^{1/k})}{k}=\operatorname{li}(n)-\sum_{k=1}^{\infty}2\ ...
-2
votes
0answers
54 views

About the concept of prime numbers [on hold]

I have developped a new approach of the concept of prime number and I generalized the concept to the reals ! Can you tell me what you think about it, please ? I have already published an article in a ...
7
votes
1answer
158 views

Prove $18080108080 \sum_{k=0}^{1560-1} 10^{10k}+1$ is prime

I saw this fact on twitter: I would like to know how one would show this number is prime. Is there an elementary way to show that this number is prime? Is there a simplified primality testing ...
6
votes
1answer
212 views

Is $7^{8}+8^{9}+9^{7}+1$ a prime? (no computer usage allowed)

Prove or disprove that $$7^{8}+8^{9}+9^{7}+1$$ is a prime number, without using a computer. I tried to transform $n^{n+1}+(n+1)^{n+2}+(n+2)^{n}+1$, unsuccessfully, no useful conclusion.
5
votes
2answers
113 views

Is this Goldbach-type problem easy to solve?

Problem: Given an odd prime number $p$, are there odd prime numbers $q$, $p'$, $q'$ such that $\{p,q\} \neq \{ p',q'\}$ and $p+q = p'+q'$ ? This comment informs that it's an obvious ...
5
votes
1answer
213 views

Identity for frequency of integers with smallest prime(n) divisor

An identity for A038110 and A038111: $$ \frac{\phi(e^{\psi(p_{n}-1)})}{e^{\psi(p_{n})}}=\frac{\prod _k^{n-1} \left(1-\frac{1}{p_k}\right)}{p_n}, $$ where $\psi(\cdot)$ is the second Chebyshev function ...
26
votes
0answers
314 views
+50

Very tight prime bounds

Is it possible that $$\left|\operatorname{li}(n)-\sum_{k=1}^{\lfloor\log(n)\rfloor}\dfrac{\pi(n^{1/k})}{k}-\log(2)-\dfrac{1}{2}\right|<\dfrac{2\sqrt{n}}{e\log(n)}?$$ Since $$ ...
0
votes
1answer
183 views

Why elements of the set can be Goldbach pairs for a given even number?

Let's take even number $100$ as an example (an example in the paper): Fom $2$ to $\sqrt{100}$ there's four primes:$\ 2,\ 3,\ 5,\ 7.\ $Let $$ \begin{align*} &A=\{n: n \in \mathbb{Z^+}, ...
4
votes
2answers
46 views

Density in $\mathbb{R}_{ +}$ of a subgroup of $\mathbb{Q}_{> 0}$?

Let $\phi : \mathbb{Q}_{>0} \to \mathbb{Z}$ be the group morphism defined by $\phi(p) = p$ for $p$ a prime number. It follows that $\phi(1)=0$, $\phi(a.b) = \phi(a)+\phi(b)$, $\phi(a^{-1}) = ...
1
vote
1answer
64 views

Is there a prime between $k$ and $\dfrac{11}{9}k$, $\forall k\ge 24$?

Given $k\in\mathbb{N}$, $k\ge 24$, is there always a prime number in the interval $\left[k,\dfrac{11}{9}k\right]$? I tried to verify this statement with the computer and it seems to hold. Is it ...
10
votes
2answers
1k views

Do 4 consecutive primes always form a polygon?

Related to this question, if 4 segments have length of 4 consecutive primes, can they always form a 4-vertex polygon? This question occurred to me out of sheer curiosity, but now I can't prove or ...
0
votes
0answers
14 views

Semiprime error margin

As an extension to this question, the plot below shows $$\pi(x)-R(x)\ \ \text{ (blue)},$$ $$\pi_{(2)}(x)-smoothed\left[ ...
2
votes
0answers
25 views

Semiprime asymptotic step function

Since $$\pi_{(2)}(x)=\sum_{i=1}^{\pi(x^{1/2})}\left(\pi\left(\dfrac{x}{\text{p}_i}\right)-i+1\right),$$ where $\pi_{(2)}(x)$ denotes the semiprimes and $\text{P}_i$ is the $i$th prime, an asymptotic ...
3
votes
1answer
68 views

Probability that two random integers have only one prime factor in common

The probability that two integers picked at random are relatively prime is known to be $1/\zeta{(2)}=6/\pi^2\approx0.607927...$. Generalizing, the probability that $n$ random integers have $\gcd=1$ is ...
0
votes
0answers
91 views

Arithmetic progression and average of two prime numbers

Let $A=(a_n : n \in \mathbb{N})$ be the sequence given by: $$ \ a_n = a_1 + (n - 1)d,\quad a_1,\ d,\ n \in \mathbb N,\quad d\gt a_1,\quad \gcd(a_1,\ d)=1. $$ For all terms of $A$ greater than $\ ...
2
votes
0answers
50 views

totally split primes in a number field

I have to show: For any number field $K$, there are infinitely many prime numbers $p \in \mathbb{N}$, that are totally split in $K$. I think have already shown (with some hints my professor gave) ...
3
votes
3answers
125 views

Number of primes less then $6000$ using $n/ \log n$

So I am trying to use this formula here and is giving me some trouble. If I just substitute $6000$ into the formula, the answer is approximately $1500$. But the number of primes under $6000$ is ...
2
votes
1answer
73 views

Infinite families of prime numbers

What interesting/useful infinite families of prime numbers are there? Right now it would be useful if I could find one with arbitrarily many 1's in its binary representation, but I am doing a larger ...
7
votes
5answers
665 views

Induction hypothesis misunderstanding and the fundamental theorem of arithmetic.

The fundamental theorem of arithmetic is made of two parts: The existence part: There exist primes such that for any natural number $j$, we can write $j$ as a product of prime numbers. The ...
2
votes
0answers
57 views

Birthday problem & primes

Let $\pi_k(n)$ be the almost prime counting function, then $\pi_k(2^kn)$ reaches a max value, since $\pi_k(2^kn)=\pi_{k+1}(2^{k+1}n)$ for large enough $k$. (eg, ...
2
votes
1answer
67 views

$\text{lcm}(1,2,3,\ldots,n)\geq 2^n$ for $n\geq 7$

I can prove that $\text{lcm}(1,2,3,\ldots,n)\geq 2^{n-1}$. Newly, i read in a paper that for $n\geq 7$ we have: $$\text{lcm}(1,2,3,\ldots,n)\geq 2^n$$ Can you prove it? (this inequality is an ...
2
votes
0answers
81 views

Questions about central polygonal numbers $1, 2, 4, 7, 11, 16, 22, 29, 37, 46,\cdots$

Formula for Central polygonal numbers is $\frac{n(n+1)}{2} + 1$, if $n=1$ or $n$ is prime, we get the new sequence $A$: 2, 4, 7, 16, 29, 67, 92, 154, 191, ... It seems that all primes either is ...
3
votes
1answer
42 views

Can a Mersenne number be a power (with exponent > 1) of a prime?

Let $n \geq 1$ and consider the (Mersenne) number $M_n = 2^n-1$. Is it possible that $M_n = p^k$ for some prime $p$ and some (necessarily odd) $k > 1$? Thanks in advance.
3
votes
0answers
33 views

Prime Triangle:: How to find the position(row and column) of prime number in a triangular arrangement

I was working on problem which asks the position of a prime number in a triangular arrangement. If we arrange the all prime up to 10^8 as shown in image we can find the row and column number of a ...
3
votes
1answer
59 views

Improving Bertrand's postulate

Recall that Bertrand's postulate states that for $n \ge 2$ there always exists a prime between $n$ and $2n$. Bertrand's postulate was proved by Chebyshev. Recall also that the harmonic series $$ 1 + ...
1
vote
0answers
43 views

Wolstenholme Number

Does Wolstenholme Numbers have perfect squares other than 1 and 49? The first few are 1, 5, 49, 205, 5269, 5369, 266681, 1077749 seems to be a complicated problem
1
vote
1answer
43 views

Proof for Carmichael theorem

if $n=p_1^{a_1}p_2^{a_2}p_3^{a_3}\dots p_r^{a_r}$ and $\lambda(n) = lcm[(p_1-1)(p_1^{a_1-1}),(p_2-1)(p_2^{a_2-1}),(p_3-1)(p_3^{a_3-1}),\dots,(p_r-1)(p_r^{a_r-1})]$ then $k^{\lambda{n}} \equiv ...
1
vote
1answer
32 views

Twin primes : prove the convergence of $ \lim_{N\sim\infty} \frac{1}{N} \sum^{N}_{p\in T} (\log(p)+\frac{1}{p})²$

let $T$ be the twin primes set : $p \in T $ if and only if $ p$ and $p+2$ are primes. Can you help me establish the convergence of : $$ \lim_{N\sim\infty} \frac{1}{N} \sum^{N}_{p\in T} ...
1
vote
1answer
71 views

Combining primes for getting primes?

I was thinking about what would happen when we combine two prime numbers $p$ and $q$ into one number $:pq:$ . Like if $p=5$ and $q=3$ , then $:pq:=53$ . Then if $p=7$ and $q= 11$ then $:pq:=711$ and ...
5
votes
1answer
178 views

Why all composite numbers have this property?

Define $f(n)=\sum\limits_{A \in S} f_{1}(n,A),\ n>2,\ n \in \mathbb{Z}$, where $S$ is the power set of $\{\frac{1}{2},\cdots ,\frac{1}{n-1}\}$. Define $\ f_1(n,\varnothing)=1,\ ...
0
votes
2answers
32 views

$\log$ approximation for $\pi(x)$

It seems that a reasonable $\log$ approximation for $\pi(x)$ can be given, where $f(y, x) := \log\left(\dfrac{\log(x)}{\log\left(e(y - \lfloor y\rfloor) + x^{1/x}(1 - y + \lfloor ...
1
vote
1answer
55 views

Finding Prime triples with $p_{n} +p_{n+1} −p_{n+2} = 1$

I was just looking at a sequence of primes and suddenly I got this thought that $p_2 +p_3 −p_4 = 1$ since $p_2 = 3, p_3 = 5, p_4 = 7$. Also for $p_3 = 5, p_4 = 7, p_5 = 11$ one has $p_3 +p_4 −p_5 = ...
1
vote
2answers
53 views

Intuitively, what separates Mersenne primes from Fermat primes?

A Mersenne prime is a prime of the form $2^n-1$. A Fermat prime is a prime of the form $2^n+1$. Despite the two being superficially very similar, it is conjectured that there are infinitely many ...
1
vote
0answers
32 views

On non-divising primes of an integer $x$

We know more about divisor than non-divisors, If we consider the sets : $$P^{1}_{x} =\left \{ p \leq x : \ p \in \mathbb{P} \right \}$$ ($\mathbb{P} $ is the primes set) $$ P^{2}_{x} =\left \{ ...
1
vote
1answer
55 views

How many natural value of n such that $n^5+2n^4+n-1$ is prime number?

From above polynomial, I can only get one value to make it prime. The value, I guess, is only one. For $n=1$, we got: $$(n^5+2n^4+n-1)= 1+2+1-1= 3 \quad\text{(prime)}$$ But, I cannot find the ...
3
votes
0answers
45 views

Solution of a equation in natural number nvolving reciprocal of prime

Let $p$ be a prime and $n$ a natural number . Solve in $\mathbb{N}$ the equation $$\sum_{k=1}^{n}\frac{1}{x^k_k}=\frac{1}{p}$$
1
vote
1answer
47 views

The most efficient method for generating new prime numbers

What is the most efficient method for generating a prime number larger than the largest known prime number, and what is the complexity of this method? Techniques considered: Mills' Constant - ...
0
votes
1answer
32 views

References for Legendre's prime-counting function

This question is about Legendre's prime-counting function, the one that can be used to calculate the exact amount of prime numbers that are less than or equal to a given number (as long as the number ...
5
votes
2answers
168 views

least common multiple $\lim\sqrt[n]{[1,2,\dotsc,n]}=e$

The least common multiple of $1,2,\dotsc,n$ is $[1,2,\dotsc,n]$, then $$\lim_{n\to\infty}\sqrt[n]{[1,2,\dotsc,n]}=e$$ we can show this by prime number theorem, but I don't know how to start I ...
3
votes
1answer
21 views

Find all $n$ so that $c_n$ $>$ $\pi(n^2)$

Find all $n$ $\in$ $\mathbb{N}$ so that $p_{c_n}$ $>$ $n^2$ where $p_n$ denotes the $n$-th prime and $c_n$, the $n$-th composite. I have tried doing the problem using The stronger version of ...
3
votes
1answer
51 views

Conjecture about prime numbers

$$\forall k\in\mathbb{N},k\ge1,\exists p:k^3\lt p\lt (k+1)^3$$ with $p$ prime number. In other words is it possible to prove that for every $k\gt1$, with $k$ integer number it exists a prime number ...
2
votes
0answers
58 views

Is this a conjecture or an already existing one??

Does the following inequality hold? $p(n)\leq 2^n,$ where $p(n)$ is the $n$th prime. If this is true then it follows that: If $p(n)=p(m)^x+p(o)^y$, then $\max[x,y] \le n$.
1
vote
1answer
67 views

Sum of reciprocals of primes for known primes.

I was reading through some old analytic number theory notes earlier and found the interesting fact that even though $\sum\frac{1}{p}$ diverges: $\sum_{\text{known primes}}\frac{1}{p} < 4$. ...
0
votes
1answer
56 views

Under assumption that $\frac{M_{n+1}}{M_n} \le 2$, what is true?

This question was hinted upon with the still open question at [1]. Let $M_n = $ A005250($n$) of the OEIS. That is to say, $M_n = p_{i+1}-p_i$, where $p_i$ is the smallest prime such that $p_{i+1} - ...
6
votes
1answer
85 views

(Easy?) consequence of the Riemann Hypothesis

I'm trying to show that the relation $\psi(x)=x+O(\sqrt{x}\log ^2 x)$ (consequence of the Riemann hypothesis) implies $\pi(x)=Li(x)+O(\sqrt{x}\log x)$, where $Li(x)=\int_2^x \frac{dt}{\log t}$. I ...
1
vote
1answer
29 views

Any simpler way to do Pollard's p-1 method?

I found calculating factorization by Pollard's p-1 method is almost impossible if use a conventional scientific calculator. For example, I am trying to factor ...
1
vote
1answer
47 views

On Newman/Zagier's proof of PNT

I have just got this paper: http://people.mpim-bonn.mpg.de/zagier/files/doi/10.2307/2975232/fulltext.pdf and I have a serious doubt: When proving that soft Tauberian theorem he explicitly uses ...
1
vote
1answer
33 views

Divisibility of sum of exponents

Consider the sequence $$r, \ ra, \ ra^2, \ ra^3, ... \ , ra^n \mod M $$ such that: $$ ra^{n+1} \equiv r \mod M$$ and $a \ne 1$ and $a,r$ are both coprime to $M$ Is it always true then that: ...
5
votes
1answer
102 views

Zhang's theorem and Polignac's conjecture

Yitang Zhang made a groundbreaking discovery when he proved that there are infinitely many pairs of prime numbers which differ by less than $70,000,000$. Zhang's theorem has been significantly ...