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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Better approximation for prime numbers [on hold]

The prime number theorem gives this formula for approximating the number of primes up to $n$: $\frac{n}{\ln(n)}$. By looking at this image from Wikipedia, I noticed that the function giving ratio ...
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Using The Riemann Zeta Functional Equation

Riemann was able to establish the following link between the Riemann zeta function and the weighted prime counting function $J(x)$. $$ln(\zeta(s))=s\int_1^\infty J(x)x^{-s-1}dx$$ Using the Mellin ...
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Why isn't this considered a formula for calculating the $n$th prime? [duplicate]

According to this math.stackexchange.com question, there is no formula for the $n$th prime. Why isn't this considered a formula for calculating it? $$ \\\text{prime}(x) = ...
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Probability number is divisible by half the square of a prime?

Let $p$ be a prime. What is the probability that a number of the form $\left \lceil \frac{p^2}{2} \right \rceil$ divides a random positive integer $n$. I have a solution that involves the Riemann-Zeta ...
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Is there a way to estimate the number of positive integers less than or equal to $n$ that have a given prime $p$ as a least prime factor

The probability that an integer $p$ divides an integer $x$ is $\dfrac{1}{p}$. From this article on almost prime numbers, the number $\pi_k(n)$ of positive integers less than or equal to $n$ with at ...
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Distribution of the sum reciprocal of primes $\le 1$

$$\frac{1}{2}+\frac{1}{3}+\frac{1}{7}+\frac{1}{43}+\cdots \le 1 $$ This is an interesting infinite summation. This is very closely resembling my other problem with has to do with the distribution of ...
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Finding Gaussian Primes along Lines in $\mathbb Z[i]$

I am trying to prove the following statement: For all positive integers $a$ does there exists a positive integer $b$ such that $a^2 + b^2$ is prime? (If so, can we provide such a $b$?) Given the ...
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Find all numbes $1\le a\le n-1$ which are prime to n and they are not witness Fermat of compositeness of n

Given the number $n=35$.Find all numbes $1\le a\le n-1$ which are prime to n and they are not witness Fermat of compositeness of n I found this problem on internet and i am trying to find a solution ...
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Find all numbers that have 30 factors and have 30 as one of their factors.

Find all numbers that have 30 factors and have 30 as one of their factors. Thank you. Note: please show way if possible.
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2answers
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Do prime numbers have prime factors?

(This is a somewhat trivial question). Do prime numbers have prime factors, i.e. itself? For example is 7 a prime factor of 7? The reason I ask this is because there is a statement in my lecture ...
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Find the smallest prime number that doesn't divide any 5-digit number whose digits are in strictly increasing order.

Find the smallest prime number that doesn't divide any 5-digit number whose digits are in strictly increasing order. I have posted an answer of my own below; any alternative solutions will also be ...
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Why does symmetry happen in reset-based random walks?

I am studying the basic concepts about random walks / brownian motion, and based on the idea of a Möbius-based walk in Wolfram's website, I wanted to try my own version of it in Python to compare it ...
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1answer
18 views

What is the summatory function of the number of (not necessarily distinct) prime factors?

In the Math World article on Merten's Constant, a related constant $B_2$ is mentioned which "appears in the summatory function of the number of (not necessarily distinct) prime factors." I am very ...
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How is the Twin Primes Constant useful? What value does it provide over Brun's Constant?

The Twin Primes Constant is: $$\prod_{p > 2 \text{ and a prime }}\left(1 - \frac{1}{(p-1)^2}\right) = 0.6601618158\ldots$$ It appears that in this case $p$ does not have to be a prime. But if ...
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How do I show that:if$p$ is prime $>5$ then $p^4-20p^2+19$ is always divisible by $180$.?

Is there someone who can show me How do i show that :If $p$ is a prime number greater than $5$ then : $$p^4-20p^2+19$$ is always divisible by $180$. Note : i think should factor $p^4-20p^2+19=$ ...
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Prime from mirror concatenation of first primes

The mirror concatenation of the first 1, 6 and 8 prime numbers with no primes being reversed is a prime ! i.e. 131175323571113 and 19171311753235711131719 are prime numbers! (beautiful primes!). After ...
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226 views

How to get this equation solved?

I came across this equation $$\sqrt a-\sqrt b=\sqrt 7-\sqrt 5$$ And you have to find the value of '$a$' and '$b$' when both of them are primes. The solution was $a=7, b=5$. Now, my question is, ...
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Proof that $2^n-1$ does not always generate primes when primes are plugged in for $n$?

Exactly what the name entails. The function $2^n-1$ I see largely tends to generate primes when $n$ is prime. However, a week ago I heard that this was horribly false. Please show me a disproof.
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Is there a function that shows n!-lcm(1,2,3…n)? [on hold]

1!-lcm(1)=0 2!-lcm(1,2)=0 3!-lcm(1,2,3)=0 4!-lcm(1,2,3,4)=12 5!-lcm(1,2,3,4,5)=60 I have not looked into this incredibly deeply, but I found that there may be some sort of connection with primes, ...
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Is there a function that can be subtracted from the sum of reciprocals of primes to make the series convergent

The gamma constant is defined by an equation where the harmonic series is subtracted by the natural logarithm: $$\gamma = \lim_{n \rightarrow \infty }\left(\sum_{k=1}^n \frac{1}{k} - \ln(n)\right)$$ ...
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Proving that $i! \mid (p-1)\cdot(p-2)\cdots(p-i+1)$ for $i < p$

Started solving this problem: $$ (a+b)^p \equiv a^p+b^p \pmod{p}$$ where $p\in\mathbb{P}$, $a,b\in\mathbb{Z} $ After a few implications I arrived to this $$ i! \mid ...
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what is the greatest integer that divides $p^4-1$ for every prime number p greater than 5

what is the greatest integer that divides $p^4-1$ for every prime number p greater than 5(this is a gre subject math problem) I think that $p^4-1=(p^2+1)(p-1)(p+1)$,so 8 must divide all the $p^4-1$ ...
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Inverse of prime counting function

The prime counting function $ \pi (x) \approx \dfrac {x} {\ln(x-1)} $. This function returns the number of primes less than $x$. Note: $x-1$ gives a better estimate than $x$. How to find $x$ given $ ...
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If Wieferich primes are finite…Then what?

I am wondering if $1093$ and $3511$ are the only Wieferich Primes, then what would it imply? (A wieferich prime is a prime satisfying the congruence $2^{p-1}\equiv 1\ mod \ p^2 $). I know of 3 cases; ...
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70 views

Are all even numbers the difference of prime powers

Does there exist an even positive integer greater than $100$ (to eliminate trivial cases) that cannot be expressed in the form: $p^2-q$ $p-q^2$ $p^2-q^2$ $p^3-q^3$ where $p$ and $q$ are primes.
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Quadratic polynomials describe the diagonal lines in the Ulam-Spiral

I'm trying to understand why is it possible to describe every diagonal line in the Ulam-Spiral with an quadratic polynomial $$2n\cdot(2n+b)+a = 4n^2 + 2nb +a$$ for $a, b \in \mathbb{N}$ and $n \in ...
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Is $\pi(n)$ a Rational Function?

Are there some two-variable polynomials $P(n,\log n)$ and $Q(n,\log n)$ which we have the bellow equation for prime counting function $\pi(n)$ for $n \in \mathbb{n}$? $$\pi(n) = \Bigl{\lfloor} ...
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Prime Zeta Function proof help: Why are these expressions not equal?

I was trying to create a formula for the Prime Zeta function and I partially succeeded except for one frustrating error. I was only able to formulate an approximation. Consider the following sum: ...
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$\pi(x)$ Proof Clarification

In a proof from a number theory book that $${\pi(x) \over x}\le {2k \over x} + {\phi(k) \over k}$$ Where $x=kl+r$ with $0 \le r\lt k $ It is stated that $$\pi(x) \le k+(l-1)\phi(k) + r \le 2k+{x\over ...
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Usefulness of prime numbers as Threading Timeouts in programming [closed]

I am a .NET programmer, founded in math. I am having an argument with a fellow programmer. When I add a Threaded Timer to the program, the interval in milliseconds I use is always a prime number. ...
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Is $u_n$ where $\left\{\begin{matrix}u_{1}=5 \\ u_{n}=\frac{2^{u_{n-1}}+1}{3} \end{matrix}\right.$ always prime?

$\left\{\begin{matrix}u_{1}=5 \\ u_{n}=\frac{2^{u_{n-1}}+1}{3} \end{matrix}\right.$ I conjecture that $u_{n}$ is prime number. But I can not prove it. So I want to know my conjecture is right or ...
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Does $\pi \left(\dfrac{x+y}{2}\right)=\pi\left(\sqrt{xy}\right)$ hold for infinitely many values of $x$ and $y$?

The problem is (assume $\pi(x)$ to be the prime-counting function), Does there exist infinitely many solutions to the equality $\pi \left(\dfrac{x+y}{2}\right)=\pi\left(\sqrt{xy}\right)$ with ...
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Is the Riemann zeta function $\zeta(s)$ exactly $\pi(x)$?

Let $\pi(x)$ denote the number of primes less than or equal to a certain x value. The prime number theorem says that $x/\log x$ (or more accurately $x/(\log x-1)$) has been the most popular method ...
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Are 7 and 49 coprime?

Or 6 and 36, 5 and 30, and things like that. They aren't, right? A co prime is a pair of numbers whose greatest common factor is 1. They (7 and 49) share 7 as well as 1. If 7 and 49 aren't co prime, ...
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What is known about the sum $\sum\frac1{p^p}$ of reciprocals of primes raised to themselves?

Consider the following series: $$\sum_{p\in\mathcal{P}}\frac{1}{p^p}$$ where $\mathcal{P}$ is the set of all prime numbers: $\mathcal{P}=\{2,3,5,7,11,13,\ldots\}$. My question is: Is this a ...
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Is it allowed to define a number system where a number has more than 1 representation?

I was just curious; is it allowed for a number system to allow more than one representation for a number? For example, if I define a number system as follows: 1st digit (from right) is worth 1. 2nd ...
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Storing a natural number as a set of its Nth prime factors, how much data is used?

Spoiler, tap to reveal. In asking the following question, I knew that each natural number could be prime factorised. However I assumed that most natural numbers would each be equal to the ...
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sextic reciprocity and divisibility question

Regarding the question if $p|(2^{2(p-1)/6}+2^{(p-1)/6}+1) $ where $p$ is a prime of the form $7\mod 8 $ That is how far I got: $2^{(p-1)/6} \mod\ p\equiv x $ if the solution of $x^6\ mod\ ...
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why only square root approach to check number is prime [closed]

Why do we use only square root approach to find a number is prime or not? why not cube root & 4rth root?
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Given a number $N$ and a prime $P$, how many numbers $\leq N$ are divisable by P but not by any smaller primes?

The following Math Exchange question deals with a similar problem: not divisible by 2,3 or 5 but divisible by 7 However, the answers given become infeasible quite quickly because the amount of ...
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how can one show that if $m$ and $n$ are co-prime, then $U_n$ and $U_m$ are also co-prime?

Given that $$U_n=\underbrace{1\cdots1}_{n\text{ times}}$$ and $n >2$, how can one show that if $m$ and $n$ are co-prime, then $U_n$ and $U_m$ are also co-prime? Because $U_m= ...
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Why is $x^{100} = 1 \mod 1000$ if $x < 1000$ and $\gcd (x,1000) = 1$?

Let $U(1000) =$ the multiplicative group of all integers less than and relative prime to $1000$. "Show that for every $x \in U(1000)$ it is true that $x^{100} = 1 \mod 1000$." Been thinking ...
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$p$ and $r$ are primes greater than $2$. $p+r$ vs $p+2r$, which could be a prime number?

For $p+2r$, a example would be $3$ and $5$. Since $6+5 = 11$, I am led to believe $p+2r$ to be the right answer. But I don't know how it works?
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Trouble with a proof: $(p^n - 1 , e)=1$ for $e\in \mathbb{N}$, p prime

I'm having trouble understanding a proof. The Lemma states: For every natural number $e$ there are infinitely many prime powers $q$ with $(q-1,e)=1$. The prove is as follows: Write $e=2^km$, m odd. ...
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What is currently the biggest prime number with no smaller undiscovered prime number? [duplicate]

Just out of curiosity, what is currently the biggest discovered prime number with no smaller undiscovered prime number?
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Does there exist a prime that is a sum of two prime power towers? [closed]

Does there exist prime number of the form $$\huge 2^{3^{5^{\,.^{.^{.\,^{p_n}}}}}} + p_n^{p_{n-1}^{\,.^{.^{.\,^{3^{2}}}}}}$$ where $p_n$ is the $n$-th prime number(and both towers are running through ...
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60 views

prove that $p(n) := n^2 + n + c$ is not prime

The question is in MIT Mathematics for CS assignments but unfortunately there is no solutions. -> I do understand that it is false if we use $n = c$ or $n = c-1$ but cannot formally write it as ...
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Is it correct to say a number $n$ is prime if $n \bmod a \neq 0$ for $2 \leq a \leq\sqrt n$?

As I was playing around with Fermat's little theorem, I came up with another method to check if numbers are prime, if the remainder of the division of $n \over a$ was not $0$ for any integer a between ...
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163 views

Showing that the Prime Number Theorem is Plausible.

I have started to work through the course notes titled "Integers, Polynomials and Finite Fields" by Kenneth Davidson to keep me busy this summer, and there is a question in here This is an ...
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Average smallest prime factors

I looked at the average smallest prime factor (ASPF) for the numbers up to N: $\text{ASPF}(N) = \frac{1}{N-1}\ \Sigma_{k=2}^N \text{SPF}(k)$ ASPF(100) = 13 ASPF(1,000) = 79 ASPR(10,000) = 578 ...