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.

learn more… | top users | synonyms

0
votes
1answer
17 views

Primality Test for Safe Primes

Is this proof acceptable ? Theorem Let $N$ be of the form $N=2p +1$ with $p$ prime , then $N$ is prime iff $N \mid 2^{2p}-1$ Proof In one direction , if $2p+1$ is a prime then by Fermat ...
14
votes
0answers
275 views

Irrationality of $\displaystyle\sum_{p\in\mathbb{P}} \frac{1}{2^{p}}$

Let $\mathbb{P}$ be the set of prime numbers, and consider $m=\displaystyle\sum_{p\in\mathbb{P}} \frac{1}{2^{p}}$. Is $m$ irrational? In the following paper, the author recalls several sufficient ...
6
votes
1answer
99 views

Prime number conjecture

It was suggested that I put my full conjecture up instead of specific examples. Here it is: Given any prime p>3, there exists c such that the following conditions hold: 1a. The quadratic equation ...
0
votes
0answers
26 views

Estimate, using the Knuth-Trabb-Pardo table, how many values of $r$ would be needed in order to factor…

Use the Knuth-Trabb-Pardo table to estimate, for the original Quadratic Sieve, with all $r \ge \sqrt{n}$, approximately how many values of $r$ would be needed in order to factor a forty-digit ...
0
votes
1answer
69 views

Are there smaller orders (cardinalities) of infinity?

I am using this source as a basis for the language to ask this question. Considering the topic of degrees of infinity, are there smaller degrees than ℵ0 (aleph null, also called ω)? ...
10
votes
9answers
989 views

A Poster About Prime Numbers [closed]

We're going to design a poster about prime numbers, which will appear in a mathematics magazine for middle school students. The poster should be both visually attractive and mathematically rich. Do ...
2
votes
1answer
39 views

Irrationality of Decimal Expansion of Primes

I've heard the proof that this number is irrational is accessible to even a novice to number theory: $\alpha = 0.2 \ 3 \ 5 \ 7 \ 11 \ 13 \ 17 \ldots$ The proof may utilize that a number is ...
0
votes
1answer
55 views

A question on the prime number theorem as presented in the following paper

In the section 2. of this paper it is written that, ...The prime number theorem ensures that we can choose $B$ as close to $1$ as we want, provided $x_0$ is sufficiently large. I think that ...
1
vote
1answer
42 views

Consecutive prime numbers

Let's assume k and n are consecutive prime numbers, $k \lt n$. An axiom: for any such $k$ and $n$, $k^2 \gt n$. This seems 'obviously' true to me, but could you please prove me wrong? Or if it's ...
0
votes
0answers
32 views

Triangular number puzzle with big numbers

Let $n_T$ be the $n^{th}$ triangular number, 1+2+3+...+n or $\sum_{i=1}^n i$ , which equals ${n(n+1) \over 2}$ . Show there exists some positive integers m and c, such that the following are true: ...
1
vote
1answer
44 views

A question about the product of primes

Let $\mathbb{P}$ be the set of all primes in the natural numbers and let $p_i \in \mathbb{P}$ be the $i$th prime, $p_1=2$. Let $m = \prod_{i=1}^n (p_i)$. How many solutions does $x^2 + x \equiv 0 ...
6
votes
1answer
259 views

Showing unique prime factorization in first-order logic?

Suppose I have the symbols $\{\neg, \rightarrow, =, <,\cdot, \leftrightarrow,\land, \lor \}$ and functions $Div(x,y)$ ($x$ divides $y$), $Prime(x)$ true if $x$ is a prime, and domain $\mathbb{N}$. ...
1
vote
1answer
55 views

To prove $\pi(x)>\dfrac x{\ln x} , \forall x \ge 17$ by elementary argument

Is there an elementary argument for proving $$\forall x \ge 17:\pi(x)>\dfrac x{\ln x} $$ ? where $\pi(x)$ is the prime counting function ....
5
votes
0answers
81 views

Primality of $n! +1$

I came across with a problem where I was required to examine primality of $n! +1$ (17! + 1 was the actual number). Although Wilson's Theorem could be manipulated for determining primality of $n! + ...
0
votes
1answer
39 views

A Generalization of Carmichael Numbers

Obviously, from Fermat's Little Theorem, the condition of $p$ being prime is equivalent to there being some number $a$ of multiplicative order $p-1$ mod $p$. Moreover, this is equivalent to saying ...
0
votes
1answer
13 views

finding A using with restriction $1 \leq a \leq 20$ in GCD

For what $1 \leq a \leq 20$ you are finding $a$ is it true that $a^m+a^n=x^2$ for positive integers $a,m,n,x.$ I did $a^m+a^n=x^2.$ $=a^m(a^{n-m}+1)=x^2$ We know that since $(a,b)=1$ since the ...
-1
votes
1answer
28 views

Asymptotic Expression for the Twin Prime Counting Function

A variation on a previous question I asked, which has garnered no responses. I'll attempt to be more lucid: Let $\pi_2(x)$ be the twin prime counting function and $\pi(x)$ be the prime counting ...
3
votes
5answers
2k views

If $p$ and $q$ are prime numbers larger than $2$, then $pq + 1 $ is never prime

I am trying to prove the following: If $p$ and $q$ are prime numbers larger than $2$, then $pq + 1 $ is never prime. Any ideas?
0
votes
1answer
94 views

Counting prime powers

The number of prime powers (exponents $\geq$ 2) up to x is given by: $x^\left(\frac12\right)+x^\left(\frac13\right)+x^\left(\frac14\right)+ $...$ =O(\sqrt x$ $lnx) $ ...
0
votes
0answers
10 views

Primality Criterion for Specific Class of Proth Numbers

Is this proof acceptable ? Theorem : Let $N = k\cdot 2^n+1$ with $n>1$ , $k<2^n$ , $3 \mid k $ , and $\begin{cases} k \equiv 3 \pmod {30} , & \text{with }n \equiv 1,2 \pmod 4 \\ k ...
-1
votes
0answers
16 views

Relationship between density of twin primes and super-primes?

One can define the twin prime counting function as $\pi_{2}(x) = \sum_{n=3}^{x} a_n$ where $a_n = [\pi(n) - \pi(n-1)][\pi(n+2) - \pi(n)]$ and the super-prime counting function as $\pi_{s}(x) = ...
1
vote
2answers
39 views

Show that if $p$ is an odd prime, show a power $p^k$ can never be a perfect number

Show that if $p$ is an odd prime, show a power $p^k$ can never be a perfect number. I am little confused about this problem, any insight?
0
votes
0answers
50 views

How can one find a million of consecutive prime numbers greater than 1 trillion? [duplicate]

I am looking for bigger prime numbers than 1 trillion. At least a million consecutive ones. Where or how can I find some?
4
votes
2answers
1k views

Where can one find a list of prime numbers?

I am looking for the biggest list of precomputed prime numbers one can find and download. Where should I look?
0
votes
2answers
19 views

Proof for divisions that in include prime number. [duplicate]

How do I prove, that if $m^2$ can be divided $p$ (where $m$ is a whole number and $p$ is a prime number) then also m can be divided by $p$?
2
votes
2answers
123 views

Prove or disprive that $n^{2}-n+17$ is prime for all integers $n$

I am looking to prove this function is always prime for all integers $n$: $$n^{2}-n+17$$ I have tested it for the first $10$ integers and it seems to work but I am not sure how to prove it form all ...
2
votes
2answers
238 views

Proof of lack of pure prime producing polynomials.

I recently encountered this following proposition: For every polynomial, there is some positive integer for which it is composite. What is the most elementary proof of this?
9
votes
6answers
687 views

Is there always a prime number between $p_n^2$ and $p_{n+1}^2$?

The following table indicates that there is a prime number p between the square of two consecutive primes. $$ \displaystyle \begin{array}{rrrr} \text{n} & p_n^2 & p_{n+1}^2 & \text{p} \\ ...
0
votes
1answer
31 views

The primes such that removing digits from the right end leaves another prime

The number 73,939,133 is prime. Keep removing a digit from the right end. Each of the remaining numbers is prime. How to find other numbers with this property?
2
votes
0answers
18 views

Does there exist a sequence $(S_i)_{i=1}^{\infty},\ S_i=\pm1$ such that $\forall i(2+S_1g_1+S_2g_2+\cdots+S_ig_i\in\Bbb P)\wedge\exists i:S_i=-1$?

Consider a sequence $(S_i)_{i=1}^{\infty},\ S_i=\pm1$ other than $\{1,1,\ldots\}$. Let $g_i=p_{i+1}-p_i$, where $p_i$ is the $i$th prime. Is it possible that for all $k\in\Bbb Z^+,\ ...
1
vote
1answer
10 views

Does there exist a positive integer $k$ such that $(g_i^k)_{i=1}^{\infty},\ g_i^k=p_{i+k}-p_i$ is non-decreasing for all sufficiently large $i$?

Does there exist a positive integer $k$ such that $(g_i^k)_{i=1}^{\infty},\ g_i^k=p_{i+k}-p_i$ is non-decreasing for all sufficiently large $i$, where $p_i$ is the $ith$ prime? $g_{i+1}^k\geq ...
4
votes
2answers
69 views

Generalisations of primes

I've read of (normal) primes, Gaussian primes and Eisenstein primes, which all uses different ways to define an integer to be a prime. For instance, $2$ factors into $1-i$ and $1+i$ for guassian ...
0
votes
0answers
136 views

Number of primes with $-1\pmod 6$ vs. Number of primes with $+1\pmod 6$

Given the following two sets: $P^-(n) = \{p \leq n : p \equiv -1\pmod 6\}$ $P^+(n) = \{p \leq n : p \equiv +1\pmod 6\}$ For example: $P^-(40) = \{5,11,17,23,29\}$ $P^+(40) = \{7,13,19,31,37\}$ ...
1
vote
0answers
150 views

Remarks on a Previous Post

Recently I have been reading this post and I have noted something significant in the fake argument. As one can easily see that the basic idea behind the argument had been to show that the sequence ...
3
votes
2answers
133 views

Simple Application of Prime Number Theorem

I was wondering if someone could possibly help me verify the following: For sufficiently large $n$ there is always a prime between $n- \sqrt{n}$ and $n$. I am not sure if this is true or not. If it ...
1
vote
2answers
51 views
0
votes
2answers
30 views

an irreducible polynomial over GF(2) is primitive over GF(2)

let $P \in F_{2} [X]$ of degree $7$, how to prove this: P is irreducible $\Leftrightarrow$ P is primitive i tried to use the mersenne prime !
2
votes
4answers
237 views

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 $2$, $5$, $10$, $17$, $28$, ...
1
vote
2answers
38 views

Prove that if p divides xy then p divides x or p divides y

I am given that the following proposition is true. (Proved in class) "Suppose that $x$, $y\in \Bbb Z$, not both zero. Then there exists $m$, $n\in\Bbb Z$ such that $$mx + ny = d$$ where $d$ is the ...
0
votes
1answer
90 views

Prime number in set $\{1,…,60\}$

How can we calculate by using the principle of inclusions and exclusions how many prime numbers are in the set $ \{1, ..., 60 \} $?
1
vote
0answers
20 views

Syndeticity and A.P.-richness of certain sets

Let $A \subset \mathbb{N}: \sum_{a \in A} (\frac{1}{a}) = \infty$; denote $\{ \alpha_1 @ \alpha_2: \alpha_1, \alpha_2 \in A \} = A @ A$, where "$@$" is any appropriate binary operator. (Note: $A$ is ...
3
votes
1answer
61 views

Given $0 < p < n$, prove there exists $n$ consecutive natural numbers such that each natural is divisible by at least $p$ distinct primes.

Given $0 < p < n$, prove there exists $n$ consecutive natural numbers such that each natural is divisible by at least $p$ distinct primes. Is there a general proof method to prove this ...
0
votes
1answer
23 views

Disproving using a Constructive Proof

I cannot find the n to prove the negation for the following: Disprove (Prove the negation) of: For every positive integer n, $3^n + 2$ is prime The way in which I have written the negation is: ...
2
votes
3answers
90 views

Finding all the values of n, such that $ \varphi (n) = 12 $ [duplicate]

I have not broken this down very far. I have come to the conclusion that there are infinitely many values for n where there exists 12 coprimes to n. Since there are infinitely many primes, and primes ...
0
votes
1answer
32 views

Uses of Mersenne primes in math

There is an international search for Mersenne primes. The project is huge. But what are the uses of Mersenne Primes in math? Do they have any other properties other than being of the form $2^n-1$?
1
vote
0answers
31 views

Fundamental Theorem of Arithmetic (Canonical) missing crucial step

I've worked long on the proof of the fundamental theorem of Arithmetic and there is only one tiny piece left I can't wrap my head around. Suppose that $$\prod_{i=1}^r p_i^{m_i} = \prod_{j=1}^s ...
4
votes
1answer
92 views

Proof of $p_n<n^2$ by Elementary Means

Is there any proof of the inequality $p_n<n^2$ (for all sufficiently large $n$) by elementary means and without using Prime Number Theorem? I searched in google but in vain. The results that I ...
12
votes
1answer
182 views

Primes in $\lfloor a^{n} \rfloor$

Motivated by the question Is there any result, that says that $\lfloor e^{n} \rfloor$ is never a prime for $n>2$?, take a real number $a>1$ and consider the sequence $\lfloor a^{n} \rfloor$. ...
3
votes
1answer
62 views

Is this number composite or prime: $2000^{2002} + 2000^{2000} + 1$?

Is this number composite or prime? $$2000^{2002} + 2000^{2000} + 1$$ I want to find an easy approach to this problem.
5
votes
1answer
41 views

Normalizer and centralizer are equivalent when $p$ is the smallest prime dividing $|G|$

Let $p$ be the smallest prime dividing $|G|$, and suppose that some $P \in \mathsf{Syl}_p(G)$ is cyclic. Prove that $N_G(P) = C_G(P)$. So I let $G=p^\alpha m$ $p$ does not divide $m$. P is cyclic, ...