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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Prime triplets and congruences

Show that if $n$, $n+2$ and $n+6$ are a prime triplet then $4320(4((n-1)!+1)+n)+361n(n+2)\equiv0\ \pmod{ (n(n+2)(n+6)}.$
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2answers
529 views

Primes approximated by eigenvalues?

Consider the matrix starting: $$\displaystyle T = -\begin{bmatrix} +1&+1&+1&+1&+1&+1&+1 \\ +1&-1&+1&-1&+1&-1&+1 \\ ...
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1answer
120 views

Is my conjecture true? : Every primorial is a superior highly regular number, and every superior highly regular number is a primorial.

I have invented two sets of positive integers: highly regular numbers and superior highly regular numbers. A positive integer $m \leq n$ is a regular of the positive integer $n$ if all prime numbers ...
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Is there a Poulet number with this condition?

Is there a Poulet number $n$ with this condition: $◎(n)=\frac{n+1}{2^x}$ or $\ ◎(n)=\frac{n-1}{2^x}, \ x \in \mathbb{N}_{\gt 0}$? (Recall that a Poulet number is a composite $n$ such that $2^n−2$ is ...
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1answer
66 views

Prove or disprove that $\forall k\in\mathbb N$ there exist tree consecutive primes such that $p_i-p_{i-1}\gt k$ and $p_{i+1}-p_{i}\gt k$

Prove or disprove that for every positive integer $k$, there exist tree consecutive primes $p_{i-1}, p_i, p_{i+1}$ such that $p_i-p_{i-1}\gt k$ and $p_{i+1}-p_{i}\gt k$. It's well known that ...
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2answers
139 views

A question about the Andrica's conjecture on the prime numbers

The Andrica's conjecture on the prime numbers states: given a couple of prime numbers $p_k$ and $p_{k+1}$ the following inequality holds: $$\sqrt{p_{k+1}}-\sqrt{p_{k}}\lt 1$$ Is it possible to show ...
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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 ...
3
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1answer
29 views

Group of numbers in residue number system of first $n$ primes

Denote $$\Pi_n = \Pi_{i=1}^n p_i$$ ie. the product of the first $n$ primes. Assume we have a number $m$. Denote $$L(m) = k \iff \Pi_{k-1} < m \le \Pi_{k}.$$ Assume also $L(1)=1.$ Now, if $L(m) = ...
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Connes trace doubt and operator $ H=xp$

i am trying to understand the paper from page 315 and on http://www.alainconnes.org/docs/bookwebfinal.pdf a) in the form of a sum of primes what does the integral $$ \int _{Q_{p}} ...
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48 views

For all prime $\ p > \ 2,\ p=2^x \cdot Ord_p(2)+1$?

For all prime $\ p\ > \ 2,\ p=2^x \cdot Ord_p(2)+1?\ $ Where $\ x \in \mathbb{Z}_{\geq 0}.\ $ Such as $\ Ord_3 (2) = 2, \ 3=2^0 \cdot 2 + 1$. Is there some way to prove this?
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1answer
31 views

Cut a piece of dough into $3$ even pieces

How could you cut a piece of dough into $3$ even pieces? Cutting it into $2$ is easy, but it's not that trivial for greater numbers. If you can cut it into $n$ pieces, you could repeat the process on ...
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0answers
28 views

Has anyone used the isomorphism with $\Bbb{N}_{\gt 0}$ as a monomial ordering?

Let $R[x_1, x_2, \dots]$ be a ring of formal polynomials in a countably $\infty$ number of indeterminates $x_i$, over a commutative ring $R$. The commutative monoid $X = \{ x^e = x_1^{e_1} x_2^{e_2} ...
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1answer
29 views

Given $N$ is the product of $m$ distinct primes, how many $K(K+1)$ are divisible by $N$?

Let $N$ be the product of two distinct primes $P_1$ and $P_2$. Let $S$ be a set of all $K<N$, such that $N|K(K+1)$. For example, if $N$ is $5\cdot7=35$ then $S=\{14,20\}$. The size of $S$ in ...
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0answers
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Subfactorial primes

So I just did some stuff and from what I can see, if y > x then !x + !y can only be prime if y = x+1 (apart from a few small exceptions near the start of the list. I don't know anything about ...
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Does every prime divide some Fibonacci number?

I am tring to show that $\forall a \in \Bbb P\; \exists n\in\Bbb N : a|F_n$, where $F$ is the fibonacci sequence defined as $\{F_n\}:F_0 = 0, F_1 = 1, F_n = F_{n-1} + F_{n-2}$ $(n=2,3,...)$. How can ...
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1answer
302 views

the sum of ascending powers of a prime can not equal the sum of ascending powers of a different prime.

I was thinking about a question that I can't prove and can't find any proof/counter example for so here it is: prove the equation: $$ \sum\limits_{i=0}^x p_1^i = \sum\limits_{i=o}^y p_2^i $$ has no ...
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2answers
70 views

Is it sufficient for a number to be a prime if it is not divisible by prime numbers smaller than it?

I am student of computer science with no knowledge of maths. To write a small algorithm I searched for the solution first. There are many but almost all of them state that continue dividing the number ...
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71 views

Show a polynomial is irreducible, ring homomorphisms, prime numbers

Let $p$ be a prime number. 1) Show that the linear transformation: $r_p:\mathbb Z[x] \to \mathbb F_p[x]$ that replaces the coefficients with the remainders of their division by $p$ is a homomorphism ...
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1answer
98 views

Is there a known function $f(n) = P_n$, where $P_n$ denotes the $n$th prime number?

Is there a known function $f:\mathbb{R}\to\mathbb{R}$, such that: The definition of $f$ does not contain the $!$ operator The definition of $f$ does not contain the $\sum$ operator The definition of ...
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Compute $ \lim_{n\to \infty}\prod_{i=1}^n B(p_i^{-2})$

Let $B(x) = \begin{pmatrix} 1 & x \\x & 1 \end{pmatrix}$, and $2=p_1<p_2<\cdots <p_n <\cdots$ primes number. Compute $$\displaystyle \lim_{n\to \infty}\prod_{i=1}^n ...
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Decimal form of irrational numbers

In the decimal form of an irrational number like: $$\pi=3.141592653589\ldots$$ Do we have all the numbers from $0$ to $9$. I verified $\pi$ and all the numbers are there. Is this true in general for ...
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4answers
463 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, ...
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1answer
33 views

Prove that there are an infinity of prime $ak+b$, $a$ and $b$ coprimes

We have to integers $a,b$. I need to show that if $a$ and $b$ are coprimes then the set of prime numbers of kind $ak+b$ is infinite. How could I show it ? I know how to do that for $4k+3$ or $4k+1$, ...
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3answers
83 views

Smallest Prime Factor - Why does this algorithm find prime numbers?

I have been looking at the problems on Project Euler and a number of them have required me to be able to find the prime factorisation of a given number. While looking for quick ways to do this, I ...
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1answer
31 views

Algorithm for checking Prime Power

Suppose we are given some arbitrary positive integer. How can we check whether the integer is a prime power? Brute force would be very inefficient in this case.
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Does there exist any integer $ n> 1$ for which $6^{2n}-25$ is prime?

I got this question on a test and I am really curious hoe you would approach it. I tried to prove stuff using the congruence laws but I didn't manage to prove anything.
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Finding an integer (if one exists) $n$ such that $n$, $n+1$, $n+2$, $n+3$, $n+4$ are all composite

I started off by thinking I would have to work $\bmod 24$ (as $24=1\cdot2\cdot3\cdot4$) But I then decided to multiply all of the terms together, and have ended up with a rather large expression. I'm ...
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1answer
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Show that $\mathbb{Q}(\zeta)$ contains one of the two numbers $\sqrt{\pm5}$ and decide which one is contained in $\mathbb{Q}(\zeta)$.

Let $\zeta$ be the 15th primitive root of unity in $\mathbb{C}$, show that $\mathbb{Q}(\zeta)$ contains one of the two numbers $\sqrt{\pm5}$ and decide which one is contained in $\mathbb{Q}(\zeta)$. ...
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1answer
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Is my proof correct regarding the non primality of $2\cdot 17^a +1$?

Today I need your help to know if the proof I have provided below is correct or not. I want to prove that there is no prime of the form $2\cdot 17^a+1$ where $a\in \mathbb N$. Now, first of all, I ...
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Finding smallest and largest prime factor of $\frac{200!}{180!}$

I originally started off by listing all the primes: $p<200$ then trying to calculate the prime factorisation of each (which I realise is a silly thing to do) I believe there must be a simpler way ...
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1answer
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How to prove if an arithmetic function is multiplicative?

I know that for an arithmetic function to be multiplicative then $f(nm)=f(n)f(m)$ for $(n,m)=1$ I have just proved that: $$f(n) = \left\{ \begin{array}{l l} 0 & \quad \text{if 10|n}\\ ...
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1answer
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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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if N is the Sum of P primes, computer program homework [closed]

Given a natural number $N$ and a positive integer $P$. I need to write a computer program that outputs if $N$ can be represented as a sum of $P$ prime numbers which need not be distinct. We need to ...
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$n$th prime bounded from above?

Let $p_n$ be the $n$th prime, $p_n\#\equiv\prod_{k=1}^{n}p_k$ (primorial), and $\sigma(n)=\sum_{d|n}^{}d$ (divisor function). Does $\text{exp}\bigg(\dfrac{\pi^2}{6 ...
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1answer
66 views

Existence of a prime partition

I'm interested in finding out whether there exists a prime partition of a given positive integer $N>1$ such that the partition has specific number of parts. For instance, as given in another ...
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67 views

Among the superior highly composite numbers, which are the most divisor dense numbers?

I’m searching for the most divisor dense natural numbers. Firstly we have the highly composite numbers: 1, 2, 4, 6, 12, 24, 36, 48, 60, 120, 180, 240, 360, 720, 840, 1260, 1680, 2520, 5040, … But ...
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2answers
51 views

$\pi(x)\leq \frac x{f(x)}$ for some unbounded function $f(x)$

Let $\pi(x)$ denote the number of primes $\le x$. Can one prove $$\pi(x)\leq \frac x{f(x)}$$ for some function $f(x)(x\gt0)$, and $f(x)$ is unbounded? Please do not refer to prime number ...
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Complexity of finding the largest prime factor of a composite number

Is finding the largest prime factor of a number computationally easier than factoring the number into powers of primes?
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Using Bertrand's Postulate

Using Bertrand's postulate which states: For every integer $n \geq 1$ there is a prime number p such that $n<p\leq 2n$ Prove that there exists infinitely many primes whose decimal expansion ...
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1answer
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Show $x^{\pi(x)} < 3^x$ using the PNT.

Using the Prime Number Theorem show that: $$x^{\pi(x)} < 3^x$$ for sufficiently large $x$. I started off by taking the $\log$ of the inequality such that: $$\log(x^{\pi(x)}) < ...
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1answer
120 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 ...
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1answer
311 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 ...
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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 ...
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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 $p_n\sim n\log(n\log(n))$?
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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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2answers
115 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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5answers
60 views

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 ...
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1answer
68 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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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 ...
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3answers
72 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.