Prime numbers are natural numbers not divisible by any smaller number other than 1. This tag is intended for questions about, related to, or involving prime numbers.

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$n$th prime & prime number theorem

Let $p_n$ be the $n$th prime. If $\pi(n)\sim \dfrac{n}{\log (n)}$ then $p_n\sim n\log n$ (Hardy 1938). A closer approximation is $\pi(n)\sim\text{Li}(n)$. Is there a similarly improved definition for ...
2
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1answer
162 views

How do we prove $p_n\sim n\log(n\log(n))$ from the Prime Number Theorem?

Let $p_n$ be the $n$th prime. Could someone please help me with the steps between $\pi(n)\sim\dfrac{n}{\log(n)}$ and $n=\pi(p_n)$, to the statement $p_n\sim n\log(n\log(n))$?
4
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1answer
122 views

Proof of Infinite Primes in the form $10^{\lceil k \log_{10}(n) \rceil }+n^{k-1}$

Let $k$ be any positive integer then how to prove that the sequence $$Q_k=10^{\lceil k \log_{10}(n) \rceil }+n^{k-1}$$ Contains infinitely many primes? It seems like because if you look at some ...
5
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1answer
69 views

Sum of Residues Modulo $p^2$.

Let $p$ be an odd prime. Prove that $$ \sum_{k = 1}^{p-1} k^{2p-1} \equiv \frac{p(p+1)}2 \pmod{p^2}$$
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5answers
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Understandng euclids theorem

Reading this Wikipedia article, it states "If q is not prime, then some prime factor p divides q" Why does some prime factor divide q? Does mean that for any number there is some prime factor p that ...
2
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2answers
117 views

Asymptotic divisor function / primorials

Let $p_n\#\equiv\prod_{k=1}^{n}p_k$ (primorial), and $\sigma(n)=\sum_{d|n}^{}d$ (divisor function). Could someone please tell me what the general asymptotic of $\dfrac{\sigma(p_n\#)}{p_n\#}$ is? It ...
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3answers
73 views

Find primes $p_1,p_2,..,p_6$ such that $1+\prod_{i=1}^{6}p_i $is not prime

Show that if$$ p_1, p_2, p_3, p_4, p_5, p_6 $$are primes, then $$1+\prod_{i=1}^{6}p_i$$ is not necessarily prime by using a specic example.
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1answer
51 views

Existence of semi-primitive primes modulo a special class of numbers

Let $p$ be a prime and $N$ be an integer. Then $p$ is called semi-primitive modulo $N$ if there exists a positive integer $j$ such that $p^j \equiv -1 \pmod{N}$. Now let $m$ be a positive integer ...
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0answers
39 views

Existence of primes $p$ such that all the prime divisors of $p+1$ divide $p-1$

This question recently came up to me in a project and is not taken from a textbook. I would like to know if any characterization of such primes is known from literature. They are seemingly rare but do ...
3
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0answers
37 views

When is $f(n)=\sum\limits_{d\mid n}\sigma(d)$ prime?

When is $f(n)=\sum\limits_{d\mid n}\sigma(d)$ prime? Note, $f$ is multiplicative and $\sigma(n)>1, \;n>1$. Therefore $f(n)$ is prime only when $n=p^\alpha$, with $p$ prime, $\alpha\geq1$. ...
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0answers
41 views

Are there such prime giving functions?

Here let us define a function $f : \mathbb{N} \rightarrow \mathbb{N}$ , such that for every $n$ , The sequence $\{f(n) ,f(n)+1 ,f(n)+2 , f(n)+3, \dots , f(n)+n\}$ contains atleast $1$ prime . Let us ...
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1answer
22 views

Finding the Best Constant in Prime Counting Function Relation

How close can we approximate the best constant $c$ such that $n^{\pi(2n)- \pi(n)} \le c^n$ for all positive integers $n$. I know that $c = 4$ works from $n^{\pi(2n)-\pi(n)} < \prod_{n < p \le ...
3
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0answers
152 views

Conjecture on OEIS A167055

OEIS A167055 Numbers n such that $12n + 5$ is prime. $0, 1, 2, 3, 4, 7, 8, 9, 11, 12, 14, 16, 19, 21,...$ are items of OEIS A167055. I conjecture that the set of the sum of every two items of this ...
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0answers
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Prime Zeta Function

Does $$\sum_{p \text{ prime}} \frac{1}{p^s} \sim \log \zeta(s) \quad \text{as} \quad s \to 1^+$$ imply $$\sum_{p \leq n} \frac{1}{p} \sim \log H_n \quad \text{as} \quad n \to \infty,$$ where $H_n$ is ...
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4answers
466 views

What is the largest prime number? [duplicate]

I want to know, what is the largest prime number? I know prime numbers are whole numbers that cannot be divided by any whole number except 1 and themselves, I also know some primes like 2, 3, 5, 7, ...
2
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1answer
79 views

A Shorter Proof of Rosser's Theorem Without Using The Prime Number Theorem

While researching on the elementary proof of Bertrand's Postulate I came to know about a theorem of Rosser's which states that $p_n$ $>$ $n$ $\text{ln}$ $n$. I have seen Rosser's original proof and ...
4
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1answer
95 views

Is this sequence monotonically decreasing?

Let $a_n = \frac{p_n - p_{n-1}}{p_n \log p_n}$ where $p_n$ denotes the $n$-th prime. Is this sequence decreasing (or decreasing after some $N$)?
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2answers
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About twin primes and their happy mothers.

Let's say that a positive integer $n$ is a happy mother if $6$ divides $n$ and $(n-1,n+1)$ is a pair of twin primes. Is the difference between two consecutive happy mothers necessarily a happy mother ...
4
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3answers
141 views

Tell whether $\dfrac{10^{91}-1}{9}$ is prime or not?

I really have no idea how to start. The only theorem considering prime numbers I know of is Fermat's little theorem and maybe its related with binomial theorem. Any help will be appreciated.
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0answers
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About a paper by Gold & Tucker (characterizing twin primes)

I've carefully looked at the questions on prime and twin prime, but the following question seems not to habe been asked before. Context: In the paper by Jeffrey F. Gold and Don H. Tucker titled A ...
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3answers
33 views

Why is $a^c-1$ composite if $a>2$ or if $c$ is composite?

Here is the original theorem from my book (A Course in Number Theory by H.E.Rose, 2nd edition): Let $a>1$ and $c>1$ be integers. The integer $a^c-1$ is composite if $a>2$ or if $c$ is ...
4
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2answers
108 views

Do prime numbers satisfy this?

Is this true that $n\log\left(\frac{p_n}{p_{n+1}}\right)$ is bounded, where $p_n$ is the $n$-th prime number?
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3answers
205 views

Does $\sum_{p\in\mathbb P}\frac {( - 1)^{[\sqrt p\,]}}{p}$ converges?

Does $$\sum_{p\in\mathbb P}\frac {( - 1)^{[\sqrt p\,]}}{p}$$ converges ? I know that the following $\sum_{p\in\mathbb P} \frac{1}{p}$ diverges, we can find proofs on Wikepedia Divergence of the ...
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0answers
26 views

Does sum over primes of $p^{-z}$ diverge for all Re(z) = 1?

Let the function q(z) of one complex variable z be the sum over all primes p of (1/p^z). I was wondering about the complex zeros of q(z) [hoping that this problem might be much easier than the same ...
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1answer
64 views

Show that $b_n > b_{n-1}$ where $\frac{a_n}{b_n}$ are the n:th harmonic number

Let $H_n=\frac{a_n}{b_n}$ where $H_n$ is a n:th harmonic number and $a_n$ and $b_n$ are coprimes. 1/ If $n$ is a prime power, show that $b_n > b_{n-1}$ 2/ Find the integer factorization of ...
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5answers
299 views

Is $n^2 + n + 1$ prime for all n?

I recently stumbled across this question in a test. Paul says that "$n^2+n+1$ is prime $\forall\:n\in \mathbb{N}$". Paul is correct, because... Paul is wrong, because... The ...
4
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1answer
59 views

Primes in an Infinite Set

Let $S$ be the infinite set of positive integers whose members can be written with no digits except $0$ and $1$ and with no more than $1988$ $1s$. Show that some integer $n$ does not divide any member ...
3
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2answers
51 views

Assume that $p$ is a prime, $a$ and $b$ are integers such that $p \mid b$ and $am+b=1$.

Assume that $p$ is a prime, $a$ and $b$ are integers such that $p \mid b$ and $am+b=1$. Prove that $x \equiv m(1+b+b^2+...+b^{k-1} \bmod {p^k}$ is the solution to $ax\equiv 1 \bmod{p^k}$. So I got ...
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2answers
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Integer-valued polynomial

Let $f(x) \in \mathbb{Q}[x]$, and suppose $f(n)$ is an integer for all large integer $n$. Prove that $f(n)$ is an integer for small positive integers $n$. I read the answer from here is the hilbert ...
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3answers
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Find values of $n$ that yield a prime number

Let $n$ be a positive integer, and $\frac{n(n+1)}{2}-1$ is a prime number. Find all possible values of n. What I have so far is this: $$\frac{n(n+1)}{2}-1=2, n=2$$ Also, $n^2+n-2\over2$ can be ...
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1answer
47 views

Prove that there are infinitely many relatively prime solutions of $x^2+y^2=z^3$

Show that for all integers k, there is a solution with $x=3k^2-1$ and $z=k^2+1$ You will need to calculate $y$ to show that there is such a solution, and show that the solution $(x,y,z)$ is ...
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0answers
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Estimating the Twin prime constant

http://numbers.computation.free.fr/Constants/Primes/twin.html it says: "This last constant occurs in some asymptotic estimations involving primes and it's interesting to observe that it may be ...
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1answer
21 views

Lemma about a prime ideal in a commutative ring with identity

I am trying to prove the Cyclotomic polynomial is irreducible over $\mathbb{Q}[x]$ for any prime $p$ using Eisenstein's Criterion. However, I would like to be more specific and prove the following ...
2
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1answer
39 views

Find all ordered triples $(x,y,z)$ of prime numbers satisfying equation $x(x+y)=z+120$

This question was from my Math Challenge II Number Theory packet, and I don't get how to do it. I know you can distribute to get $x^2+xy=z+120$, and $x^2+xy-z=120$, but that's as far as I got. Can ...
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4answers
175 views

Is there a conjecture with maximal prime gaps

Define $M_n$ to be the $n$th maximal gap between primes. That is, $M_1=1$ thanks to $3-2=1$; $M_2=2$ thanks to $5-3=2$; $M_3=4$ thanks to $11-7=4$; and in general, $M_n = p_{i+1}-p_i$, where $p_i$ is ...
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3answers
81 views

Pi and the sum of reciprocals of primes?

So I know that $$\sum_{\underset{\Large p\; prime}{p=1}}^{\infty}\frac{1}{p}$$ blows up. But doing some fun on mathematica I found out that when the sum isn't infinite, it was so close to $3$ and I ...
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2answers
24 views

solving this equation using prime numbers

Solve in $\mathbb{Z}$ the following equation: $3^x$+$3^y$=$738$, using prime numbers concept and decomposition in prime factors... I noticed that the above equation is symmetrical to $x$ and $y$, ...
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1answer
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how to solve this equation using certain concepts

Solve in $\mathbb{Z}$ the following equation: $x^6$ + $3x^3$ + $1$ = $y^4$, using, if it's possible, prime numbers & decomposition in prime factors concepts... Thanks for your time!
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Corollary to the Prime Number Theorem [duplicate]

Could someone please explain how to prove : $p_n \sim n\log n$ By using the Prime Number Theorem and letting $x = p_n$ I know that $n \log p_n \sim p_n$ and I also know that $n < p_n$ so $\log n ...
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1answer
69 views

Where did the constant with $\vartheta(x)<1.01624x$

In: Rosser, J. Barkley; Schoenfeld, Lowell (1962). "Approximate formulas for some functions of prime numbers.". Illinois J. Math. 6: 64–94. Theorem 9. $\vartheta(x)< 1.01624 x$ for $0 < x$. ...
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1answer
60 views

What's the Shannon entropy of the prime numbers?

Here's a note that calculates it as 1. Do you know of any other calculations? http://www.math-math.com/2014/05/shannon-entropy-shannon-entropy-of.html
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2answers
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Does $\frac{8k-1}{4}$ belongs to $\mathbb{Z}$?

Does $\frac{8k-1}{4}$ belongs to $\mathbb{Z}$ for some $k\in \mathbb{Z}$ ? or we can prove that this never belongs to $\mathbb{Z}$ ?
3
votes
4answers
228 views

A congruence involving prime numbers

This congruence appears in a textbook I'm reading anf it left the proof to the reader, however I cannot find my way around it. $$(a+b)^ p \equiv a^p+b^p \pmod p\text{ when $p$ prime and ...
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1answer
24 views

A series that has prime numbers as a element

I was looking at the following sequence 3,5,7,9,11,13,15,17,19,21,23,... The terms are given by $ a_n=n^2-(n-1)^2 $. When I expanded the sequence I noticed that it contained all the prime numbers ...
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2answers
74 views

State of art of prime numbers distribution [closed]

I was reading some questions about prime numbers posted in latest days and a question came to my mind: What is the state of art of the research into prime numbers distribution? I read then ...
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2answers
32 views

Proving a composite number is product of primes for a set

Let $A=\{4n+1:n\in\mathbb{N}\}=\{1,5,9,13,17,\dots\}$. We call a number $\alpha $ $\text{A-prime}$ if it doesn't have any divisors in $A$ aside from $1$ and $\alpha$, we define ...
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1answer
44 views

question about the forms of prime numbers

I was thinking about primes earlier and I thought of a hypothesis that I have been unable to prove. I was wondering whether it was a known theorem and whether anyone knows a proof or can prove (or ...
2
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2answers
43 views

Formula for the number of solutions of the congruence equation $xy-wz=0$ over $\mathbb{Z}_p$?

The equation $xy-wz=0$ has 10 solutions over $\mathbb{Z}_2$ and 33 solutions over $\mathbb{Z}_3$ (e.g. $x=y=2 \land w=z=1$ is one of the solutions). Is there any formula for the number of solutions ...
5
votes
1answer
312 views

Prove that there exists an $m$ such that for any $n>m$ there exists at least one prime between $c_n$ and $n$

Let $c_n$ be the $n$-th composite. Then the problem is to prove that- $\pi(c_n)-\pi(n)>0$ $\forall n>m$ I have tried to progress in the problem using an elementary approach. So far I have ...
5
votes
2answers
317 views

Does there exist a prime that is only consecutive digits starting from 1?

This is a problem I came up with the other day, and have absolutely no clue how to solve. The problem is: does there exist a number in the set $K$ that is prime, where $K$ is defined to be the set of ...