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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Existence of at least one prime for all sequences in the family of sequences

Prove or disprove that for a fixed $n \in N$, there exists at least one prime among the integers of the form $2^{k}n+2^k-1$ for an arbitrary $k \in N$.
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130 views

$9n^2-4$ only generates one prime? Why?

Instead of doing the work I was supposed to be doing, I played around with some numbers, and I noticed that for $n\in\mathbb{N}$, $9n^2-4$ only seems to generate a prime for $n=1$. Can anyone ...
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4answers
112 views

Is at least one of $6k + 1$ or $6k-1$ prime?

We know that any prime number ( $> 2,3$) can be written in the form $6k+1$ or $6k-1$. Is it necessary that at least one of $6k+1$ or $6k-1$ is a prime number ?
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Q: Understanding Answer of 2012 AMC 8 - #18

The problem is: "What is the smallest positive integer that is neither prime nor square and that has no prime factor less than 50?". The solution for this problem goes like this: "Since the integer ...
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Congruence modulo primes or in a polynomial ring over ${\rm GF}(2)$

Let $p, q$ be primes. Then the linear congruence $$ap \equiv r\pmod q$$ can be solved for $a\in\mathbb Z$ and will have a unique solution for each value of $r$ such that $0\leqslant a<q$. Am I ...
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Prove that 1 is the only “common” divisor of the integers n and n+2

Let n be any odd integer. Prove that 1 is the only "common" divisor of the integers n and n+2. I think you have to find gcd(n, n+2) and say that since n odd then then n+2 will also be odd. Thus n + ...
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Fast check of safe primes or Sophie Germain primes

If $p=2q +1$ with $p,q$ prime then $p$ is called safe prime and $q$ is a Sophie Germain prime. I want a faster algorithm for a safe prime test than doing two primality checks for $p$ and $q$. In ...
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Prove $18080108080 \sum_{k=0}^{1560-1} 10^{10k}+1$ is prime

I saw this fact on twitter: I would like to know how one would show this number is prime. Is there an elementary way to show that this number is prime? Is there a simplified primality testing ...
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1answer
83 views

Prime number upper bound

I am reading some written notes about a proof I do not understand, maybe some informations are missing. The result that has to be proved is the following: if $p_n$ is the $n$-th prime number, ...
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How I could transform this into product over primes :$s_p$= $\frac{1}{2^2-1}+\frac{1}{2^3-1}+…\frac{1}{2^p-1}$?

1)Can I transforme this sum into product OVER primes:$s_p$= $\frac{1}{2^2-1}+\frac{1}{2^3-1}+....\frac{1}{2^p-1}$ ? Note : p is prime number and ${2^p-1}$ is prime 2)I would be interest to know ...
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Largest $k$ such that $(p-k) = \lceil \sqrt{(p-2k) p} \rceil$

Assume $p \in \mathbb P.$ Assume $0<p-2k<p$ and the next square larger than $p(p-2k)$ is $(p-k)^2$. It is trivial to show that $p(p-2k)+k^2$ is a square. Simply $p(p-2k)+k^2 = (p-k)^2.$ ...
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268 views

Is $7^{8}+8^{9}+9^{7}+1$ a prime? (no computer usage allowed)

Prove or disprove that $$7^{8}+8^{9}+9^{7}+1$$ is a prime number, without using a computer. I tried to transform $n^{n+1}+(n+1)^{n+2}+(n+2)^{n}+1$, unsuccessfully, no useful conclusion.
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2answers
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The $i^{th}$ prime in a given ring R

When I say that $p_1=2$, I mean that the first prime in the standard ring of integers $(\mathbb{Z},*,+)$ is $2$. I was wondering whether the notion of ordering the primes like this can be generalized ...
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120 views

Is this Goldbach-type problem easy to solve?

Problem: Given an odd prime number $p$, are there odd prime numbers $q$, $p'$, $q'$ such that $\{p,q\} \neq \{ p',q'\}$ and $p+q = p'+q'$ ? This comment informs that it's an obvious ...
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101 views

Given $N$, what is the next prime $p$?

Certain data structures in programming related to collections operate in an optimal way if they have prime number of elements. This means if a program (programmer) requires $N$ (any natural number) ...
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374 views

Identity for frequency of integers with smallest prime(n) divisor

An identity for A038110 and A038111: $$ \frac{\phi(e^{\psi(p_{n}-1)})}{e^{\psi(p_{n})}}=\frac{\prod _k^{n-1} \left(1-\frac{1}{p_k}\right)}{p_n}, $$ where $\psi(\cdot)$ is the second Chebyshev function ...
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1answer
205 views

Is $2013^{2014}+2014^{2015}+2015^{2013}+1$ a prime? (usage of a computer not allowed)

Prove or disprove: $$2013^{2014}+2014^{2015}+2015^{2013}+1$$ is a prime number, without using a computer. I tried to transform the expression $n^{n+1}+(n+1)^{n+2}+(n+2)^{n}+1$, but couldn't reach ...
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0answers
143 views

Interchanging limits with the prime counting function

How does one justify that $$\lim_{s \to 1} \lim_{x \to \infty} \frac{\pi(x)}{x^s} = \lim_{x \to \infty} \lim_{s \to 1} \frac{\pi(x)}{x^s}, \quad s > 1,$$ without using the fact that the primes have ...
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1answer
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Extending $f(p^k)$ where $p$ is prime

If we have a function $f(x)$, for which we know that $f(p^k)=(p^s+1)^k p^{sk}$ where $p$ is prime, $k$ is a real number, and $s$ is a constant, how do we find $f(x)$? My try: let $k=\log_p(x)$, so ...
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Very tight prime bounds

Is it possible that $$\left|\operatorname{li}(n)-\sum_{k=1}^{\lfloor\log(n)\rfloor}\dfrac{\pi(n^{1/k})}{k}-\log(2)-\dfrac{1}{2}\right|<\dfrac{2\sqrt{n}}{e\log(n)}?$$ Since $$ ...
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What is the mean prime power?

I found this definition in a book and I did not understand the meaning of it There exists a field of order q if and only if q is a prime power (i.e., $q = p^r$]) with p prime and r ∈ N. Moreover, if q ...
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193 views

Why elements of the set can be Goldbach pairs for a given even number?

Let's take even number $100$ as an example (an example in the paper): Fom $2$ to $\sqrt{100}$ there's four primes:$\ 2,\ 3,\ 5,\ 7.\ $Let $$ \begin{align*} &A=\{n: n \in \mathbb{Z^+}, ...
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2answers
118 views

$n^2-79n+1601$ always a prime?

I am struggling with proving or disproving this: $n^2-79n+1601$ is a prime for all natural numbers $n$ (except multiples of $1601$). This somehow has a relation to Stanislaw Ulam spiral. What ...
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1answer
76 views

Given a prime p and an integer N, find the number of integers n such that 1≤n≤N and order(n!) is divisible by p

We are given a prime number $\leq 10^{18}$ and an integer N $(\leq N\leq 10^{18})$ how to find the number of integers lying in the range $1\leq n\leq N$ for which the order(n!) is a multiple of p? ...
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1answer
41 views

What's the best software for primality tests of huge numbers? (check if an integer is prime or not)

I just read an article about huge prime numbers (some with more than 10millions digits!) that are discovered using software that check if an integer is prime or not (primality test sofwares). What is ...
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Density in $\mathbb{R}_{ +}$ of a subgroup of $\mathbb{Q}_{> 0}$?

Let $\phi : \mathbb{Q}_{>0} \to \mathbb{Z}$ be the group morphism defined by $\phi(p) = p$ for $p$ a prime number. It follows that $\phi(1)=0$, $\phi(a.b) = \phi(a)+\phi(b)$, $\phi(a^{-1}) = ...
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Is there a prime between $k$ and $\dfrac{11}{9}k$, $\forall k\ge 24$?

Given $k\in\mathbb{N}$, $k\ge 24$, is there always a prime number in the interval $\left[k,\dfrac{11}{9}k\right]$? I tried to verify this statement with the computer and it seems to hold. Is it ...
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Do 4 consecutive primes always form a polygon?

Related to this question, if 4 segments have length of 4 consecutive primes, can they always form a 4-vertex polygon? This question occurred to me out of sheer curiosity, but now I can't prove or ...
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1answer
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Characterizing the primes which don't divide any Pell-Lucas number(s)

For integer $n$, let $P_n$ be a Pell number, and $Q_n$ its companion. Is there a characterization of the prime numbers $p$ which don't divide any $Q_n$? By brute-force search, I found that this ...
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RSA aloghorithm - stuck on d

I'm sorry in advance if this sort of question has been posted before. I couldn't find it. I'm clearly an idiot, and I clearly need help, so here I am. I have a homework assignment which overall is ...
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Semiprime error margin

As an extension to this question, the plot below shows $$\pi(x)-R(x)\ \ \text{ (blue)},$$ $$\pi_{(2)}(x)-smoothed\left[ ...
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Semiprime asymptotic step function

Since $$\pi_{(2)}(x)=\sum_{i=1}^{\pi(x^{1/2})}\left(\pi\left(\dfrac{x}{\text{p}_i}\right)-i+1\right),$$ where $\pi_{(2)}(x)$ denotes the semiprimes and $\text{P}_i$ is the $i$th prime, an asymptotic ...
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Sets of Prime Numbers Generated By an Irreducible Monic Polynomial

Given a non-constant integral irreducible monic polynomial $f(x)$, the prime factors of its value at integers $x\in\mathbb{N}$ forms a set $\mathcal{P}(f)$. Is it possible that ...
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1answer
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Probability that two random integers have only one prime factor in common

The probability that two integers picked at random are relatively prime is known to be $1/\zeta{(2)}=6/\pi^2\approx0.607927...$. Generalizing, the probability that $n$ random integers have $\gcd=1$ is ...
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Arithmetic progression and average of two prime numbers

Let $A=(a_n : n \in \mathbb{N})$ be the sequence given by: $$ \ a_n = a_1 + (n - 1)d,\quad a_1,\ d,\ n \in \mathbb N,\quad d\gt a_1,\quad \gcd(a_1,\ d)=1. $$ For all terms of $A$ greater than $\ ...
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totally split primes in a number field

I have to show: For any number field $K$, there are infinitely many prime numbers $p \in \mathbb{N}$, that are totally split in $K$. I think have already shown (with some hints my professor gave) ...
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1answer
155 views

An integral and $\pi(n)$

Are there polynomials $P,Q\in \mathbb{R}[x]$ satisfying : $$\int_{0}^{\log n}\frac{P(x)}{Q(x)}\,\mathrm{d}x=\frac{n}{\pi(n)}\quad \text{ for infinitely many }n\in \mathbb{N}$$ Here $\pi(n)$ is the ...
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Number of primes less then $6000$ using $n/ \log n$

So I am trying to use this formula here and is giving me some trouble. If I just substitute $6000$ into the formula, the answer is approximately $1500$. But the number of primes under $6000$ is ...
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1answer
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Infinite families of prime numbers

What interesting/useful infinite families of prime numbers are there? Right now it would be useful if I could find one with arbitrarily many 1's in its binary representation, but I am doing a larger ...
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An Inequality Involving Prime Numbers

Let $p_i$ be the $i^{th}$ prime number. It seems as though the following inequality is true for all positive integers $m$ and real numbers $x>1$: ...
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Is $n! + 1$ often a prime?

Related to this question, I wonder how often $n!+1$ is a prime? There is a related OEIS sequence A002981, however, nothing is said if the sequence is finite or not... or anything in that sense...
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Question regardles primes and the fundamental theorem of arithmetic

I have been reading through my book of practice proofs and came across this particular question which has stumped me. $p$ and $q$ are primes. Prove $\forall p \in \mathbb{Z}, \forall k \in ...
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Resources about infinite primes of form $n^2 + 1$

Where can one find existing work on the following problem? Prove there are infinitely many primes of the form $n^2 + 1$. Resources about related work would also be appreciated.
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Induction hypothesis misunderstanding and the fundamental theorem of arithmetic.

The fundamental theorem of arithmetic is made of two parts: The existence part: There exist primes such that for any natural number $j$, we can write $j$ as a product of prime numbers. The ...
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If $n = 51! +1$, Then find no of primes among $n+1,n+2,\ldots, n+50$

If $n = 51! +1$, Then find no of primes among $n+1,n+2,\ldots, n+50$ Really speaking, I don't have any clue ...
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Birthday problem & primes

Let $\pi_k(n)$ be the almost prime counting function, then $\pi_k(2^kn)$ reaches a max value, since $\pi_k(2^kn)=\pi_{k+1}(2^{k+1}n)$ for large enough $k$. (eg, ...
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1answer
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$\text{lcm}(1,2,3,\ldots,n)\geq 2^n$ for $n\geq 7$

I can prove that $\text{lcm}(1,2,3,\ldots,n)\geq 2^{n-1}$. Newly, i read in a paper that for $n\geq 7$ we have: $$\text{lcm}(1,2,3,\ldots,n)\geq 2^n$$ Can you prove it? (this inequality is an ...
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Questions about central polygonal numbers $1, 2, 4, 7, 11, 16, 22, 29, 37, 46,\cdots$

Formula for Central polygonal numbers is $\frac{n(n+1)}{2} + 1$, if $n=1$ or $n$ is prime, we get the new sequence $A$: 2, 4, 7, 16, 29, 67, 92, 154, 191, ... It seems that all primes either is ...
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1answer
51 views

Can a Mersenne number be a power (with exponent > 1) of a prime?

Let $n \geq 1$ and consider the (Mersenne) number $M_n = 2^n-1$. Is it possible that $M_n = p^k$ for some prime $p$ and some (necessarily odd) $k > 1$? Thanks in advance.
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1answer
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AP term multiple of prime number

I am having this equation : (a+(n-1)d)%p=0 Here a and d can go upto 10^18 and p is prime number upto 10^9 . How to find the least value of n here? Example : If ...