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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Count numbers with prime digit

Given a number N I need to find the count of the numbers that have atleast one prime digit (2,3,5 or 7) in it. Now N can be upto 10^18.What is the best approach to solve this problem. Example : Let ...
2
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1answer
33 views

Finding the lowest number (or an upper bound to the lowest number) not congruent to a set of moduli

Note: if finding x is not possible, an upper bound, where there must be at least one number less than said number which is not congruent to the set, would be helpful. The set: For my purposes, the ...
2
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1answer
46 views

Is $\sum_{n=1}^N e^{2 \pi p_n z i}$ bounded for irrational $z$?

Let $p_n$ be the $n$th prime number. If $z$ is irrational real, is it known whether the partial sums $\sum_{n=1}^N e^{2 \pi p_n z i}$ are bounded? It seems the partial sums are unbounded if $z$ is ...
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105 views

Fastest way to find if a given number is prime

Given a random number, what would be the quickest possible way of finding out whether it was prime? Obviously, one could just iterate through the number in order to see if it was divisible by ...
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194 views

Prove modular equivalence related to carmichael function.

Statement if $n=p_1p_2p_3\cdots p_r$ (prime numbers are distinct) and $2 \le a \le n-1$ then prove that $$a^{k\lambda} \equiv \left(1+\sum_{p_i\mid\gcd(a,n)}\left(\dfrac{n}{p_i} \right) ...
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374 views

Consecutive Prime Gap Sum (Amateur)

List of the first fifty prime gaps: 1, 2, 2, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 6, 6, 2, 6, 4, 2, 6, 4, 6, 8, 4, 2, 4, 2, 4, 14, 4, 6, 2, 10, 2, 6, 6, 4, 6, 6, 2, 10, 2, 4, 2, 12, 12, 4, 2, 4. My ...
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39 views

Asymptotic behaviour of $\prod_{p \leq x} (1 + 4/(3p) + C p^{-3/2})$

I'm reading Montgomery and Vaughan and in it they state quite simply \begin{equation} \prod_{p \leq x} \left(1 + \frac{4}{3p} + \frac{C}{p^{3/2}} \right) \ll (\log x)^{4/3} \end{equation} as $x ...
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1answer
51 views

Is there a 10-digit emirp?

Does a 10-digit emirp exist? Unfortunately, the lists of emirps I could find on the Web are quite small and my programming skills aren't good enough to write a program to check all the primes up to ...
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3answers
106 views

$6n+1$ and $6n-1$ prime format [duplicate]

I recently stumbled upon a fact that all prime numbers past $3$ are of the form either $6n-1$ or $6n+1$. Is it true? at least for numbers less than $10^9$. And does it cover all primes?
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What percent of numbers are primes? [duplicate]

I understand that there are infinitely many primes and (obviously) infinitely many integers, but is there any way to calculate the total percentage of integers that are primes? Thanks
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38 views

Infinite Wilson Prime proof

An article I read recently about Wilson Primes stated that, while 5, 13, and 563 are the only known terms, there is an infinite number of Wilson Primes. I was wondering if someone could verify this ...
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1answer
21 views

Two positive integers and coprime relationship [closed]

Let $a$ and $b$ be two positive integers. If $2^a - 1$ and $2^{b-1}$ are coprime then $2^b - 1$ and $2^{a-1}$ are also coprime? Please kindly advise.
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1answer
48 views

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 ...
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2answers
60 views

Diophantine: $px^2+2=y^2$ where $p\in \mathbb{P}$

Solve the Diophantine Equation: $px^2+2=y^2$, where $p$ is a prime number and $x,y$ integers. I tried this for ages but didn't get anywhere, but I don't know any advanced machinery since I am only in ...
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1answer
72 views

What is $\limsup\limits_{n\rightarrow\infty}\left\{\frac{p_{m+1}}{p_m}|m\in \mathbb{N},m\geq n\right\} $?

What is $$\limsup\limits_{n\rightarrow\infty}\left\{\frac{p_{m+1}}{p_m}\middle|m\in \mathbb{N},m\geq n\right\} = ?$$ where $p_i$ is i'th prime number. We know that this limsup exists because of ...
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2answers
183 views

Prime Number Sum Sequence (Amateur)

SOLVED: This is false Beginning with 3, add the next consecutive prime (2) and then take that sum (5) and add that to next consecutive prime (3) to get (8), and so on... $$ 3 + 2 = 5 $$ $$ 5 + 3 = 8 ...
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3answers
80 views

Number theory divisibility - simple way to prove this is prime?

Suppose that $y$ is a positive integer, and $z$ is the largest factor of $y$ such that $z<y$, then let $x=y/z$. Prove that $x$ must be a prime number. Is there a simple way to solve this? It ...
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120 views

Property of set of prime numbers

let $\{p_1,p_2,p_2,\cdots ,p_r\}$ be the set of $r$($\ge2$) pair wise distinct prime numbers i.e.., $(i\ne j \implies p_i \ne p_j)$ for all $1\le i,j\le r$ ${Statement}$ : For any such ...
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Sorting of prime gaps

Let $g_i $ be the $i^{th}$ prime gap $p_{i+1}-p_i.$ If we re-arrange the sequence $ (g_{n,i})_{i=1}^n$ so that for any finite $n$ the gaps are arranged from smallest to largest we have a new sequence ...
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50 views

Number of prime factors of Mersenne numbers

Let $p$ be a prime and let $M_p = 2^p-1$. Is it known whether the number of prime factors of $M_p$ is unbounded above as $p \to \infty$? Also do the probabilities estimating the chance that $M_p$ is ...
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Is any elementary proof important (beside Selberg's work) ?

Is any elementary proof important (beside Selberg's work) ? Plus, why is the elementary proof of prime number theory by Selberg so important ? Selberg was awarded the Field medal is mainly because ...
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1answer
33 views

Conditions for which $(\sum p_i)(\sum \frac{1}{p_i})$ over an arbitrary $i$ for a set of primes $\{p_i\}$ is unique?

I am looking for conditions (if any are needed beyond properties of primes) for which $(\sum p_i)(\sum \frac{1}{p_i})$ over an arbitrary $i$ for a set of primes $\{p_i\}$ is unique in that there is ...
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1answer
33 views

Generalizing primality to other operations

(By "number" below I always mean an element of $\mathbb{Z}^+\setminus $$\left\{1\right \}$.) We all know that a number $p$ is prime iff it cannot be represented as $ab$ for any two numbers $a$ and ...
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1answer
225 views

Can Fermat's little theorem be used to list primes?

I was reading about Fermat's little Theorem, which states that if p is prime, then for any integer a, $a^p-a$ would be a multiple of p. So, I started wondeing if this could be used to determine ...
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218 views

Unusual behavior of 210 and 199 regarding prime numbers

Adding 210 to 199 over and over again, you get 8 more primes that can be arranged together into a 3x3 magic square: 1669 199 1249 619 1039 1459 829 1879 409 Is there any other pairs of numbers ...
3
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1answer
95 views

Golden Ratio of Primes (Amateur)

Unable to find information elsewhere, so I'll try here. What two consecutive primes are closest to producing the Golden Ratio? Or two of any Primes? Has this been determined? Thanks!
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1answer
95 views

Computing infinite product over primes

How can I compute $$ \prod_p \left(1+\frac{k}{p}\right)\exp(-k/p) $$ where $0<k<e$ and the product is over all primes $p$? Background L. G. Sathe proved [1] that there are $$ ...
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Dividing a product of different primes by another prime

A relatively straight forward question. If I were to multiple any amount of different prime numbers together say 7*3*11, is it possible to divide the product by a single other prime number say 23 and ...
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3answers
62 views

Smallest prime factor of a Mersenne number

The Mersenne numbers $M_n$ are integers of the form $2^n-1$, where $n$ is a positive integer. In the case when $n$ is a prime, are there any results known on the smallest prime factor, $p_n$, of ...
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1answer
35 views

Given a sequence, construct a function whose integral is equal to the sum of the sequence

Let $P_n$ be the sequence of prime numbers, where $P_0=2$. Given $m\in\mathbb{N}$, how can we construct $f(x)$ such that: $\displaystyle\forall{0}\leq{i}\leq{m}:f(i)=P_i$ ...
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1answer
42 views

A slightly various form of Dirichlet's theorem on arithmetic progressions

Are there infinitely many primes of the form $2n(n+1)+1$?
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Are there infinitely many prime numbers in $a_n=\frac{7\times 10^n-1}{3}$?

In the array $a_n=\frac{7\times 10^n-1}{3}$, are there infinitely many primes? (when $n={7+16k},a_n$ is divisible by $17$, so there are infinitely numbers not prime)
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1answer
46 views

A different way to solve Chinese remainder theorem

I'm doing my homework about Chinese remainder theorem $x = a_1(\mod n_1)$ $x = a_2(\mod n_2)$ As I know, x can be found by using: $$x=\{\sum_{i=1}^na_iN_i(N_i^{-1}(\mod n_i)) \}(\mod N)$$ with ...
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4answers
176 views

The fastest way to count prime number that smaller or equal N

I want to count all prime numbers that existing in N but I don't know how to count. Can any one tell me how to count prime numbers that are smaller than or equal to N in mathematics formal?
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1answer
92 views

Possible Prime Sum Pattern (Amateur)

Disclaimer: I’m an amateur, and have no advanced knowledge of math, so please forgive my ignorance as I’m just curious to know if I’ve stumbled upon something or not. Prime Numbers: 2, 3, 5, 7, 11, ...
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179 views

Does $\sum\limits_{n=1}^{\infty}\frac{1}{P_n\ln(P_n)}$ converge to the golden ratio?

The sum $\displaystyle\sum\limits_{n=2}^{\infty}\frac{1}{n\ln(n)}$ does not converge. But the sum $\displaystyle\sum\limits_{n=1}^{\infty}\frac{1}{P_n\ln(P_n)}$ where $P_n$ denotes the $n$th prime ...
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conjecture about prime numbers and distance between them

is there a name for this conjecture? Conjecture: given $p_n$ a prime number sequence where $p_1=2,p_2=3,\cdots$, then for all $n\in\mathbb{N}^*$ and $k\in\mathbb{N}$, holds that $\displaystyle ...
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2answers
35 views

Proof that if $k$ is the highest factor of any positive integer $n$ such that $k<n$, then $n/k$ is prime

It's straightforward to say that when $n$ is prime, $k=1$ since $k$ must be less than $n$. For the case where $n$ is not prime, I thought proving that the lowest factor of $n$ is prime would be the ...
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Elementary proof there are infinitely many primes of the form $4n+1$

My attempt: $4n+1$ is odd. Thus its decomposition must not contain $2$. Every odd number is either of the form $4k-1$ or $4m+1$. $(4m+1)(4k-1)$ is never of the form $4n+1$. So $4n+1$ has factors ...
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Why is $y^{x-1}-1$ divisible by $x$?

I wanted to know if there is a way to prove that $y^{x-1}-1$ is divisible by $x$. Where $x$ is a prime number and is not equal to $y$, and $y$ is any positive whole number besides $1$. For example, ...
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Sequence of primes by concatenating digits in a given base.

Given a base, $b$ is there is a sequence $\lbrace a_n\rbrace_{n\geq 0}$ where $a_k \in \lbrace 1,2\cdots, b-1\rbrace$so that the sequence: $$b_n:= \sum_{k=0}^n a_kb^k$$ is a sequence of primes ...
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1answer
41 views

Digit wise modulo for calculating power function for very very large positive integers

I am writing a code to calculate $P^Q$ where $P$, $Q$ are positive integers which can have number of digits up to $100000$. I want the result as $r = P^Q \pmod{10^9+7}$, where $10^9+7$ is a prime ...
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133 views

How do you go about finding a 12 digit prime number?

How do you go about finding a 12 digit prime number?
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1answer
102 views

Meaning of Rays in Polar Plot of Prime Numbers

I recently began experimenting with gnuplot and I quickly made an interesting discovery. I plotted all of the prime numbers beneath 1 million in polar coordinates such that for every prime $p$, ...
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1answer
101 views

Why is $p_n\sim\sum_{k=1}^{n}\log(p_k)$?

Why is $$ p_n\sim\sum_{k=1}^{n}\log(p_k) $$ where $p_n$ is the $n$th prime? In addition, is it true that $$ n\log\left(\dfrac{\sum_{k=1}^{n}\log(k)}{\log(\log(n))}\right) $$ is a better ...
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Fast algorithm for generating consecutive primes larger than N

I'm looking for a fast algorithm to generate primes larger than a random 4096 bit number $N$. I know about the Sieve of Atkin, but AFAIK it can only be used to find all primes up to a certain limit. ...
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63 views

Cramér's Model - “The Prime Numbers and Their Distribution” - Part 4

Following a previous question (here you'll find an introduction): A paper by Maier which refutes Cramer's Model suggests we should replace the heuristic "$\Bbb P(n\in\mathcal P)=1/\log n$" with ...
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Cramér's Model - “The Prime Numbers and Their Distribution” - Part 3

Following a previous question (here you'll find an introduction): The book states that almost surely $$\pi_S(x+y)-\pi_S(x)=\mathrm{li}(x+y)-\mathrm{li}(x)+O(\sqrt y)$$ as soon as $y/(\log ...
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40 views

Cramér's Model - “The Prime Numbers and Their Distribution” - Part 2

Following a previous question (here you'll find an introduction): The book states that using the convergence of the binomial distribution towards the Poisson distribution, it's easy to show that ...
2
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1answer
71 views

Finding an $n$ such that $n^2 \equiv -1 \mod p$

What is an efficient algorithm to find the first number $n$ such that $n^2 \equiv -1 \mod p$ for a prime $p$, if such an $n$ exists? Is there anything better than the brute-force approach up to $p-1 ...