Questions tagged [prime-numbers]

Prime numbers are natural numbers greater than 1 not divisible by any smaller number other than 1. This tag is intended for questions about, related to, or involving prime numbers.

Filter by
Sorted by
Tagged with
0
votes
0answers
7 views

Are there any extension of primes for decimal numbers?

Mathematicians does things which seems to be impossible. (An example would be the factorial function. Its definition was only defined for positive integers but then mathematicians gave a definition ...
1
vote
0answers
15 views

Does the mean of $\log{\text{gpf}(n)}/\log{n}$ for the first $n$ naturals have a lower bound?

Does $$\frac{1}{n-1}\sum_{i=2}^{n}{\frac{\log{\text{gpf}(i)}}{\log{i}}}$$ have a lower bound as $n\rightarrow\infty$? Here, $n\in\mathbb N$ with $n>1$, and gpf returns the greatest prime factor of ...
2
votes
1answer
43 views

Regarding Prime Numbers & Integrals

Recently, I was working on Prime Numbers and here's what I found: If $$\prod_{N=3}^{n-2} \sin \frac{\pi.(n+N)}{N} = \mathcal{P}$$ then $n$ is Prime only if $\mathcal{P} \neq 0 \ \ ; \ n\geq5 $...
0
votes
1answer
33 views

Proving that something occurs infinitely often [closed]

I was just wondering what techniques exist for proving that something occurs infinitely often? Specifically I am thinking about: what functions $f(n)$, $n$ defined on the natural numbers, yields prime ...
0
votes
0answers
17 views

Finding base from modular exponentiation

Assume that $y = m^e \pmod{n}$, where $n$ is the product of two primes and $e$ is prime. Also, for $c, c' \in \{0, \dots e-1 \}$ and $r, r' \in \{0, \dots, n-1 \}$ we have: \begin{align} r^e &= hy^...
0
votes
0answers
19 views

Can one extract $x$ from these known modular expressions?

Assume $x$ is unknown and constant, but the following quantities are/can be known: $g^x \pmod{p}$ $f(x, m) = (m, a, b) = (m, (x+h) \pmod{p-1}, g^h \pmod{p})$ where $p$ is a prime, $m$ positive ...
-2
votes
0answers
27 views

For consecutive primes p1 and p2, is it true that (p1,p2+1)=1? That is, they are coprime. P1 is sufficiently large. [closed]

For p1 and p2 consecutive primes prove that (p1,p2+1)=1. That is, p1 and p2+1 are coprime. This is not true for small primes.
0
votes
0answers
18 views

What are some elementary bounds for Mertens' Third Theorem?

In particular, I am looking for $A$ and $B$ such that, for all $x>s$, $$\frac{e^{-\gamma}}{\log x} B< \prod_{ p \leq x} \frac{p-1}{p}<\frac{e^{-\gamma}}{\log x}A.$$ Rosser provides both ...
0
votes
3answers
52 views

Explain the difficulty of find primes with the same last digit

Refer to https://oeis.org/A340800 to notice that the number of primes between two primes having the same last digit is increasing as the primes themselves increase. Is there an explanation for this? ...
0
votes
1answer
23 views

Prove that each multiple of 4 plus 3 is either prime or divisible by any $n\equiv3(\mod 4)$

I've been looking at a couple of questions in revision for an upcoming exam this year, and I've half solved a question, except for this bit. I need to prove that for every multiple of 4, plus 3 (e.g: ...
2
votes
1answer
71 views

Primes congruent to 1 modulo n

I was wondering whether the following statement is true, and if so, how to prove it: for all $n\in\mathbb{N}$, there exist prime numbers $p$ and $q$ such that $p\equiv1$ (mod $n$) and $q\equiv-1$ (...
0
votes
1answer
32 views

Let $n$ be prime, for which $k \in \mathbb{N}$ is the map $\phi_k$ a group isomorphism.

Problem: Let $k,n \in \mathbb{N}$ We define the following group homomorphism. $$\phi_k: (\mathbb{Z}/n\mathbb{Z}) \to (\mathbb{Z}/n\mathbb{Z}), [a] \mapsto [ka]$$ Let $n$ be prime. For which $k \in \...
2
votes
1answer
120 views

On the greatest prime factor of $p^2+1$, $p\in\mathbb{P}$

Are there infinitely many pairs $(p,q)\in\mathbb{P}^2$ such that $p>q$ and $p\mid q^2+1$? This is a very interesting. There are many methods for bounding the greatest prime factor of $n^2+1$, but ...
0
votes
0answers
13 views

Complete residue system mod p composed only by prime number [duplicate]

Let $m$ be an integer. The set $\{r_1, r_2, \dots, r_s\}$ is a complete residue system module $m$ if $r_i \not\equiv r_j (\text{mod } m)$ for all $i \neq j$ and for all $n \in \mathbb{N}$, there is ...
0
votes
0answers
39 views

For what value of p are these three numbers prime?

Given that $p$, $p+10$ and $p+14$ are prime numbers, find $p$. I know it's a pretty easy question and the answer is $3$, which I was able to find by trial and error, but what to do if the question is ...
1
vote
1answer
23 views

Natural numbers to prime exponents

I am currently teaching a class for middle school students who are interested in mathematics and we were talking about Hilbert's Hotel and how you could fit $\mathbb{N}\times\mathbb{N}$ guests into ...
0
votes
4answers
54 views

Prove that a number is the product of two primes under certain conditions.

Suppose $p$ be the smallest prime dividing $n \in \mathbb{Z}^+$. Prove that if $n$ is not a perfect square and that $p<n<p^3$, then $n$ must be the product of two primes. Clearly, the smallest $...
0
votes
0answers
22 views

Deterministic sinusoidal function for primality?

According to Wilsons`s Theorem in a modification by Leibniz, we have that (p – 2)! ≡ 1 (mod p) for all p > 2. Then, we should also have that the term (quotient q) is a positive integer, while, if ...
4
votes
2answers
179 views

Can all naturals can be expressed as sums of prime powers of $\{3,5,7,13\}$?

Let $S$ be a set containing all terms of form $p^k$, where $p\in\{3,5,7,13\}$ paired with all nonnegative integers $k$. The question is whether every natural number can be expressed as the sum of some ...
2
votes
0answers
79 views

Is every even integer the sum or difference of two primes?

The sum bit is Goldbach's conjecture, which is open. The difference bit seems to be open as well (e.g. according to this source). What if we allow the alternative? More precisely, has the following (...
0
votes
0answers
28 views

What is a(n) for n+1 prime for the sequence(link in body)? [closed]

I gave formula for n+1 composite in the sequence which was easier. Any suggestions for n+1 prime? There is upper bound for a(n) for n+1 prime which is about half of sqrt(n+1). Link- http://oeis.org/...
0
votes
1answer
30 views

Finding roots of polynomial $f(x)$ in $\mathbb{Z}_p = \mathbb{Z}/p\mathbb{Z}$, where $p$ is prime.

I am not quite sure how to approach such a problem. In particular, I am wondering how to find the roots of the following polynomial in $\mathbb{Z}_p = \mathbb{Z}/p\mathbb{Z}$: $f(x) = x^{2p^2}- \...
3
votes
2answers
176 views

Why $[\mathbb{Q}(\sqrt[n]{p}, \sqrt[m]{q}):\mathbb{Q}] = nm$?

As title says, I want to prove that $\mathbb{Q}(\sqrt[n]{p}, \sqrt[m]{q})$ is degree $nm$ extension of $\mathbb{Q}$ when $p \neq q$ are distinct primes. By Eisenstein's criterion, $x^{n} - p$ is ...
0
votes
2answers
38 views

Finishing of basic proof of Wilson's theorem

I've been trying to simplify my proof of Wilson's theorem so that I can easily understand it, but I'm running into some trouble. I reached the point where I proved $(p-1)! \equiv -(p-2)! \mod p$, but ...
0
votes
2answers
29 views

$\mathbb{Z}_p$ with $p$ prime is integral domain [duplicate]

I was trying to solve some exercise from my abstract algebra course, and at some point, I assumed $\mathbb{Z}/2\mathbb{Z} \cong \mathbb{Z}_2$ is an integral domain, because I thought to remember that $...
5
votes
2answers
80 views

A polynomial divisible by primes of special form

$\textbf{Problem:}$Let $p$ be a prime number of the form $9k + 1$. Show that there exists an integer n such that $p | n^3 - 3n + 1$. $\textbf{Source:}$Here My only idea was that $n^3-3n+1$ might ...
5
votes
1answer
57 views

Prove that there are infinitely many primes using $Q=p_{1}\cdot p_{2}\cdot \ldots \cdot p_{m}$ and $R=p_{m+1}\cdot p_{m+2}\cdot \ldots \cdot p_{n}$

In an exercise I am supposed to prove the infinitude of the primes using the following: Let $p_{1}, p_{2}, \cdots , p_{n}$ be the first $n$ primes and $m\in \mathbb{Z}$ with $1<m<n$. Let $Q$ be ...
3
votes
4answers
100 views

Show that $\frac{46!}{48}+1$ is not a power of 47

The original exercise was: Prove that there is no non-negative integers such that $m!+48=48(m+1)^n$ I could type something, but I've stopped at the moment I had to prove that that $\frac{46!}{48}+1$ ...
-1
votes
1answer
45 views

Is it possible to prove Infinite Primes using Picks Theorem?

Lemma: Triangles containing exactly the same number of grid points (excluding boundary), then $n$ is prime. (see Pick's Theorem and Primes.) Claim: There exists infinite such triangles, so ...
1
vote
1answer
43 views

I want an operation that keeps digits in order and result in a year number

I remember a math teacher wishing us a good year with a mail title that was something like: 101 = (1 + 2 + 3)* 4 - 5*(6 - 7) + 8*9 Except the result was 2006. Is there a website that can find a number ...
1
vote
0answers
14 views

Permutation or change of coordinates such that $ax+by+cz=1$ holds with the consideration $c>0$.

We say that three positive integers $x,y,z$ are coprime if there exist three integers $a,b,c$ such that $$ax+by+cz=1$$ Here $a,b,c$ have different signs, i.e., at least one of them is negative. I am ...
5
votes
0answers
70 views

How many solutions for $\,\sum_{k=1}^{n}p_k=p_m\cdot p_{m+1}\,$?

I ask for which pairs $(m,n)$ is satisfied $$\sum_{k=1}^{n}p_k=p_m\cdot p_{m+1}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(1)$$ where $p_k$ is the $k$-th prime. Up to $n=10^7$ I have found only the solution $(4,...
2
votes
1answer
46 views

Number of integers at most $x$ with exactly two distinct prime factors

I wish to find an asymptotic for the number of integers not exceeding $x$ with exactly two distinct prime factors. Here is a starting point: Throughout $p$ and $q$ are primes. We are interested in $\...
1
vote
0answers
65 views

On Andrica's conjecture.

We have Andirca's conjecture as $\sqrt{p_{n+1}}-\sqrt{p_{n}}<1$. I have no clue if this is a proof. I was just messing around. (And plus, my methods are probably too elementary). Denote $g(n)$ as ...
0
votes
1answer
36 views

Find all $x$ such that $|4x^2 - 12x - 27|$ is prime

Find all integers $x$ such that $|4x^2 - 12x - 27|$ is prime. I first factored $|4x^2 - 12x - 27|$ as $(2x+3)(2x-9),$ but I was unsure where to go from there.
1
vote
1answer
31 views

Showing that for any composite binary number $b > 1$, $bb$ is also composite.

Say $b$ is the binary form of a composite number $d > 1$, I want to prove that $bb$ (concatenation of $b$ with itself) is also a composite number. My approach: Given a binary number $b$, $bb$ is ...
2
votes
2answers
198 views

Conjecture about prime powers

Given an odd prime power $q$. Then there exists exactly two positive integers $r_1,r_2$ such that $r_i<q-1$ and $8qr_i+1$ is a perfect square. And it holds that $r_1+r_2=q$. Tested for $q<100$ ...
1
vote
1answer
65 views

Primes of the form $p=X^2+3Y^2$

I'm trying to work in $\mathbb{Z}\left[\frac{1+\sqrt{-3}}{2}\right]$ to show that an odd prime $p\in\mathbb{Z}$, $p\neq 3$ is of the form $p=X^2+3Y^2$ if and only if $p\equiv1$ mod $3$. The hint is to ...
0
votes
2answers
71 views

Do there exists $p$ and $q$ primes for which $p^3+1=2q^2$?

Do there exists $p$ and $q$ primes for which $p^3+1=2q^2$?
0
votes
0answers
17 views

Possible extensions of the triangular structures in Gilbreath's conjecture to tetrahedronic structures.

Gilbreath's conjecture produces an triangular matrix of witch the most elements are either $0$, or $2$. When using different colors for those two elements, very interesting triangular structured trees ...
4
votes
2answers
108 views

Finding joker numbers

Let $n$ be an integer, $q(n)$ be the smallest prime number which divides $n$ and $r(n)$ be the biggest prime number less than or equal to $\sqrt{n}$. We say that $n$ is joker if $q(n)=r(n)$. Except $8$...
1
vote
1answer
24 views

Calculate the failures of the cramer model

Using $\displaystyle\mathfrak{G}(\mathcal{H})=\prod_p\left(1-\frac{|v_p|}{p}\right)\left(1-\frac{1}{p}\right)^{-k}$ it is concluded that $\displaystyle \mathfrak{G}(7,11,13,17,19,23)\neq 0$ my ...
1
vote
1answer
100 views

Finding the number of pairs (a,b) such that gcd(a,b,n)=1

This is question 6E from the 2019 Cambridge Mathematics Tripos paper 1A (which can be accessed at https://www.maths.cam.ac.uk/undergrad/pastpapers/files/2019/paperia_4_2019.pdf). "Let $n\geq 2$ ...
1
vote
0answers
64 views

Residue of powers of a prime modulus another prime.

Given different primes a and p we know by Fermat's little theorem that $$a^{p-1} \equiv 1 \pmod{p}. $$ In the case of a = 2, a sequence for $$ 2^{p-i} \equiv j_i \pmod{p}$$ can be written as $$ f(...
0
votes
1answer
27 views

How to prove these pseudo-random number generators don't repeat until running through the entire set?

I just asked this question to find an algorithm for a pseudo-random number generator that generates 100% unique numbers for a bounded set of inputs. There are two answers I've seen. First is this one, ...
0
votes
1answer
86 views

A new(?) expression for the prime counting function

I recently found the expression $$\pi(n)=\sum_{k=1}^n\left\lfloor 2^{m(k)}\right\rfloor,\tag1 \label{eq1}$$ where $$m(k)=2k-1-\sum_{j=1}^{k}\gcd(j,k).$$ I found this by designing a characteristic ...
0
votes
2answers
74 views

Identifying a prime number: Why only check for factors up to $\sqrt{N}$? [duplicate]

My math professor told the class that For a positive integer $N$, if $N$ is not divisible by prime less than $|\sqrt{N}|$, then $N$ is prime. For example, for $139$, $12>|\sqrt{139}|>11$. But ...
0
votes
1answer
51 views

What does $p≡3\mod4$ mean, and how do you find a prime matching this pattern?

I am reading How to Generate a Sequence of Unique Random Integers where it says: What does that $p≡3\mod4$ mean? How do I find an 8-bit prime, and a 16-bit prime match that $p≡3\mod4$ statement?
2
votes
0answers
35 views

Bands in ratios of consecutive prime numbers

Does anyone know any information on the bands appearing in ratios of consecutive prime numbers? For example, if any analytic explicit or recursive formulas exist per band. In other terms, how do I ...
2
votes
3answers
82 views

Another way to prove that $81^{3^{n}}+4^{2n+1}$ is not prime for any positive integer $n$

The question is to prove that $81^{3^{n}}+4^{2n+1}$ is not prime for any positive integer $n$. The easy way to solve this is by taking the last digit of each number: $81^{3^{n}}$ ends in $1$ because $...

1
2 3 4 5
215