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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Do Prime Numbers have a Structure or do they sprout out Randomly among positive Integers? [duplicate]

Since the Order of Sequence of the Prime Numbers has not been found, it seems that all famous Mathematicians have opted for the random appearance of Primes.
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The myth of no prime formula?

Terence Tao claims: For instance, we have an exact formula for the $n$th square number - it is $n^2$ - but we do not have an exact formula for the $n$th prime number $p_n$! "God may not play ...
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Square in Interval of Primes

Denote by $a_n$ the sum of the first $n$ primes. Prove that there is a perfect square between $a_n$ and $a_{n+1}$, inclusive, for all $n$. The first few sums of primes are $2$, $5$, $10$, $17$, $28$, ...
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Proving an inequality on the sum of $\log$ of primes.

Let $S(x)=\sum_{p\leq x} \ln(p)$ where $\sum_{p\leq x}$ denotes a summation over the positive prime numbers that are $\leq x$ Prove that $\forall n \in \mathbb N, S(2n+1)-S(n+1)\leq ...
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Show inequality for the number of different prime factors

We consider the function $k(n) $ that represents the number of the different prime factors of $n$.We want to show that for $n>2$ $$k(n) \leq \frac{\log n}{\log \log n}(1+O((\log \log n)^{-1})) ...
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Half prime numbers?

I am wondering if there is a term for a number which is only divisible by its square root, one and itself? For examle $25$ can be divided by $1, 5$ and $25$. And $169$ with $1, 13$ and $169$. I am ...
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New mathematical constant formed by continued fraction with prime numbers?

Notational convention: $$\bigoplus_{k=0}^{\infty}a_k=a_0+\cfrac{1}{a_1+\cfrac{1}{a_2+\cfrac{1}{a_3+\cfrac{1}{\ddots}}}}$$ Let $$ P:=\bigoplus_{k=1}^{\infty}p_k$$ where $p_k$ is the k-th prime ...
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Upper bound number of distinct prime factors

I want to prove that if $\omega (n)$ is the number of distinct prime factors of $n$ for $n>2$ we have $\omega (n) \leq \frac{\ln n}{\ln \ln n} + O(\frac{\ln n}{(\ln \ln n)^2})$. How can I do this? ...
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Relation between Galois theory and Fermat primes

I am curious about a possible relation between Galois theory and Fermat primes. There is a general solution to any polynomial equation of degree less than or equal to $4$. The only Fermat primes (of ...
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Prime factorization of 2^(16) - 1

Trying to show the the prime factorization of $$2^{16}-1$$ without a calculator. I know that $2^{16} - 1$ yields the prime numbers$$3*5*17*257$$ because I calculated $2^{16}-1$ on my calculator ...
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Let $p \geq 5$ and prime. Show $p^2 + 2$ is divisible by three.

I know I have to use the division algorithm to put into the form $p^2 + 2 = 3q + r$ but everything I've tried after that has lead me to a dead end. I've mainly been trying to show $r=0$ or to make the ...
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1answer
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Prime number theorem lemma: prove that $\psi(x)\sim\pi(x)\log(x)$

I'm trying to follow the proof in Wikipedia that the PNT is equivalent to the assertion $\psi(x)\sim x$, by proving that $\psi(x)\sim\pi(x)\log x$, which it claims is a very simple proof. One ...
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0answers
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Mapping set of integers to irrational numbers.

Mapping Integers to Irrationals..maybe even primes? Hi i'm an undergrad currently working on a research project. I recently thought of a question that I believe would help me greatly,but have ...
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6answers
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If $p > 5$ is a prime number, then the last digit of $p^4-1$ is $0$.

If $p > 5$ is a prime number, then the last digit of $p^4-1$ is $0$ (ex.: $7^4-1=2400$). How do I prove this?
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6answers
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Proof that every number has at least one prime factor

Prove that for $ n \geq 2$, n has at least one prime factor. I'm trying to use induction. For n = 2, 2 = 1 x 2. For n > 2, n = n x 1, where 1 is a prime factor. Is this sufficient to prove the ...
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1answer
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Bijection between Natural numbers and Infinite Cartesian product of Natural numbers?

Consider a function $f(n): \mathbb{N} \rightarrow \mathbb{N} \times \mathbb{N} \times ...$ mapping each number $n$ to the set of exponents to raise each prime number $p$ to in order to obtain $n$. For ...
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1answer
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Length in time to find the longest range of primes between 2 and a 13 million character digit?

I am trying to run a program that tells me how many prime numbers there are in a range of numbers. I run it in intervals of 10,000 to 100,000. How long would the program take to determine all the ...
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Prime numbers that fits in a specific pattern

Any series $\displaystyle \sum_{k=0}^{\infty}a_k2^{-k}$, where $a_k\in\{0,1\}$, converges to some $x\in[0,2]$ and since the sequence $a_n$ is unique for each $x\in[0,2]$ there is an bijection between ...
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As$\ n \to \infty$, how does a product over the primes less than$\ p_n$ equal the same product over the primes less than$\ n$? [duplicate]

How is$$\ \lim_{x\to \infty} \log \log x \prod_{i< \log x} \frac{p_i -1}{p_i}= \\ \lim_{x\to \infty} \log \log x \prod_{p < \log x}_{p prime} \frac{p-1}{p}$$?
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Quantum Prime Root Matrix and The Periodic Table of Elements [closed]

How does the sequential harmonic order of the Prime Roots Matrix synchronize with the quantum prime root frequencies of the orbital electrons and electronic configuration of all the harmonic elements ...
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0answers
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How to show that $\sum_p \int_{p^m}^\infty f(x) dx = \int_0^\infty \pi(x^{1/m}) f(x) dx$

How do you show that for some function $f(x)$, $$\sum_p \int_{p^m}^\infty f(x) dx = \int_0^\infty \pi(x^{1/m}) f(x) dx$$ where the sum on left is taken over the set of all prime numbers $p$ and ...
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Minimal distance between coprimes

For a natural number $K$, I want to choose $n$ pairwise coprime numbers all of which are bigger than $K$ such that the distance $d$ between the smallest and the largest one is minimal. For example, ...
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Upper bound for number of primes in an interval

Let $S(x,y)$ be the number of primes $p$ in $(x, x + y]$ such that also $p + 6$ and $p + 12$ are primes. I know that $$ T(x, y) \leq 48 c \frac{y}{\log^3 y} \left( 1 + O \left ( \frac{\log \log ...
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Summation of Legendre symbol

Let $\chi_{2,q}$ be the real Dirichlet character modulo a prime $q>2$, which is not the principal one (the so-called Legendre symbol). Is it true that $$ \sum_{n=0}^{+\infty} ...
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Largest prime factor of a number

In Project Euler problem 3, where we have to find the largest prime factor of a number, one of the solution i came across is ...
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Are Mersenne prime exponents always odd?

I have been researching Mersenne primes so I can write a program that finds them. A Mersenne prime looks like $2^n-1$. When calculating them, I have noticed that the $n$ value always appears to be ...
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3answers
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Let$\ p_n$ be the$\ n$-th prime. Is$\ \lim_{n\to\infty} \log \log n \prod_{i=1}^{\lfloor \log n \rfloor} \frac{p_i-1}{p_i}>0$?

I'm less than a novice in analysis, I don't even know how to approach this. Thanks in advance.
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Let$\ p_n$ be the$\ n$-th prime. Can you give me a proof for$\ \prod_{i=1}^\infty \frac{p_i-1}{p_i}=P\approx \frac{1}{11.0453}$?

I found$\ \prod_{i=1}^\infty \frac{p_i}{p_i-1}\approx 11.0453$ on Wolfram|Alpha. Moreover, writing a paper, should one provide a proof or it is trivial? Thanks in advance.
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Divergence of a series containing primes

Is there an easy proof showing that the series $1/p$, where $p$ changes over prime numbers, is divergent?
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p-adic numbers and GCD

Given two numbers $a,b \in \mathbb{Z}$, how do we prove that the $p$-adic number of $\gcd(a,b)$ is the same as the minimum for the $p$-adic number of $a$ and the $p$-adic number of $b$? Does this ...
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Summation of non-principal real Dirichlet character

Let $q > 3$ be a prime and $$ S_q := \sum_{k=1}^{q-1} \chi_{2,q} (k) \, k, $$ where $\chi_{2,q}$ is the real Dirichlet character modulo $q$ which is not the principal one. I have to prove that ...
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Divisor function convolution

I need some help to prove that $$ (d*d)(p^k) = \frac{(k+3)(k+2)(k+1)}{6} \qquad \forall p \in \mathcal{P},\quad \forall k \in \mathbb{N}, $$ where $d$ is the divisor function and $\mathcal{P}$ the set ...
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Foundational proof for Mersenne primes

I know how to prove that, if $2^n-1$ is prime and $n>1$, then $n$ is prime. But how do we prove that, if $a^n-1$ is prime and $n>1$, then $a$ must equal 2?
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To solve for $x,y,n$ in non-negative integers , $\dfrac{x!+y!}{n!}=p^n$ , $p$ a given prime

Let $p$ be a given prime , then how do we find non-negative integers $(x,y,n)$ $\space$ , such that $\dfrac{x!+y!}{n!}=p^n$ ?
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1answer
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Is there a prime number ending with the natural number $n$

if $n$ not is divisible by 2 or 5? Example: given 813075843967837637675737563754361301, there is a prime 20813075843967837637675737563754361301 or given ...
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Fermat number factor probability

I found a question I couldn't solve: What is the probability that $2^{2^{12}}+1$ has a prime factor of $70$ digits? I found this problem hard as many number has more small prime factors than large ...
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Can you prove this formula for computing prime numbers is correct?

Sometime ago I discovered the following function for computing primes: $$ Q(x)=\text{frac} \left (\cfrac{\Gamma(x)}{x} \right )\cfrac{x^2}{x-1}= \begin{cases} x & \small \text{if $x$ is prime} ...
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Prime number distribution theory for dummies

For the distribution of prime numbers there is a hypothesis which predicts the possible positions of prime numbers called Riemann hypothesis ...
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If $x$ and $y$ are positive integers then $\frac{(2x)!(2y)!}{x!y!(x+y)!}$ is an integer

If $x$ and $y$ are positive integers then $\frac{(2x)!(2y)!}{x!y!(x+y)!}$ is an integer I have to show that the proposition above is true for any $x,y\in\mathbb{Z^+}$ by means of Legendre's ...
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How amount of Emirp Primes depends on the base of numeral system

There was a problem on searching for primes which, if their decimal notation is reverted, yield another primes, like 37 => 73 or ...
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How many terms are required to get $D$ digits of Riemann zeta prime function?

How many terms are required to get $D$ digits of Riemann zeta prime function $\zeta_p(s) = \sum_p \frac{1}{p^s}$? Sebah & Gourdon mentions that finding $\zeta_p(2)$ to 20 digits by using $\sum_p ...
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Prime number generation - speed comparison

"Efficient prime number generating" leads to some algorithms being displayed as "fast". Up to PG7.8 which does takes 65786 seconds to generate the prime numbers > ...
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Asymptotic sum of the squares of the first n primes [closed]

I know there is an asymptotic formula for the sum of the squares of the first n primes, but I have been unable to find it.
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Greatest common divisor problem involving $a^p+b^p$ [closed]

Let $\gcd(a,b)=1$ for some $a,b\ \epsilon \ \mathbb{N}$. Prove that for any odd prime p: $$\gcd\left(a+b,\frac{a^p+b^p}{a+b}\right)=1,~~~~ \text{or} ~~~p.$$
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Could you give a definition of what is a superior highly composite number using only words?

I know very well what is a superior highly composite number, but I would like to see how could we (roughly) define what is a superior highly composite number using only words (using no equations and ...
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Inequality with prime numbers

I found exercise in my book for number theory that I can't resolve. How do you show that $$p_n < e^{1+n}$$ where $p_n$ is $n$-th prime number?
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system of congruence - my approach

We have: $$k^3 + l^3 \equiv 0 \pmod{17}\\ k^2 + l^2 \equiv 0 \pmod{17} $$ And I get: $$k = 17n+r_k\\ l = 17m+r_l$$ And I analyzed possible rests respect to system of congruences. My result is: $$ ...
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Difference between two (not consecutive) primes

I am searching for an lower bound on the difference between the $(n+k)$-th and $n$-th prime number in terms of $k$. I have something like this in mind (conjecture): Let $(p_k)_k$ denote the ...
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Why does Euclid write “Prime numbers are more than any assigned multitude of prime numbers.”

In Euclid's Elements Book XI proposition 20 (http://aleph0.clarku.edu/~djoyce/java/elements/bookIX/propIX20.html), Euclid proves that: Prime numbers are more than any assigned multitude of prime ...