# 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.

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### 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 ...
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### 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 ...
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### 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 $\mathcal{P} \neq 0 \ \ ; \ n\geq5$...
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### 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 ...
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### 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^...
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### 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 ...
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### 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.
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### 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 ...
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### 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? ...
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### 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: ...
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### 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 $q\equiv-1$ (...
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### 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 ...
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### 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 ...
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### 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 (...
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### 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/...
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### 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 ...
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### 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 ...
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### 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 $\frac{46!}{48}+1$ ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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.
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### 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 ...
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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 $q<100$ ...
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### 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 ...
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### Do there exists $p$ and $q$ primes for which $p^3+1=2q^2$?

Do there exists $p$ and $q$ primes for which $p^3+1=2q^2$?
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### 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 ...
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### 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 $8$...
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### 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 ...
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### 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 $n\geq 2$ ...
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### 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 ...
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### 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?
### 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 \$...