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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Concatenating the first semiprimes to get a semiprime

The first semiprime numbers are $4,6,9,10,14,15,...$ once alternate them in order from the first semiprime, we see that $46, 469, 469101415$ are also semiprimes(!). After this the largest I've found ...
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1answer
70 views

Missing prime numbers ending in 7. [closed]

Every tenth prime number $29, 71, 113,173... $Up to the 26th such number ($1657 $a Cuban number) ends in a $1, 3 or 9;$ but none end in$ 7.$ What is the probability of this? Also there are three ...
0
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1answer
59 views

Something related to carmichael numbers.

$a^{n - 1} = 1 \bmod n$ for any prime $n$ and any $a$ prime to $n$. Yet there exists composite $m$ such that $a^{m-1} = 1 \bmod m $ for all $a$ relatively prime to $m$; $m$ being a Carmichael number, ...
37
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3answers
1k views

Small primes attract large primes

$$ \begin{align} 1100 & = 2\times2\times5\times5\times11 \\ 1101 & =3\times 367 \\ 1102 & =2\times19\times29 \\ 1103 & =1103 \\ 1104 & = 2\times2\times2\times2\times ...
1
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1answer
52 views

On an estimation of binomial coefficient

On page 14 of the book 'Proofs from THE BOOK', there is an estimation presented as: $$\binom{2n}{n}\le \prod_{p\le \sqrt{2n}}\ 2n. \prod_{\sqrt{2n}<p\le \frac{2}{3}n}\ p. \prod_{n<p\le 2n}\ p, ...
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Lifting quadratic residues

Let $p$ be an odd prime. Show that if $q$ is a quadratic residue modulo $p^x$ for some $x > 0$, then $q$ is a quadratic residue modulo $p^{x+1}$. We have $x^2=q \pmod {p^x}$, and $x^2-q=m ...
8
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3answers
74 views

Does the sum of reciprocals of primes congruent to $1 \mod{4}$ diverge?

Let $P$ be the set of primes $p$ greater than $3$ such that $p\equiv1 \pmod{4}$. Does the following sum converge or diverge? $$ \sum_{p\in P}\frac{1}{p} $$
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202 views

The effect of roots of Dirichlet's $\beta$ function condenses to $\frac12\left(1+ie^{i2\pi\frac{p}4}\right)$

With the help of Raymond Manzoni and Greg Martin I was able to derive an explicit formula for the number of primes of the form $4n+3$ in terms of (sums of) sums of Riemann's $R$ functions over roots ...
3
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1answer
91 views

Intersection between the sums of the first positive integers, primes and non primes

Conjecture : $$\left\{\sum\limits_{\begin{array}{c}k=1\\k\in\mathbb{Z}\end{array}}^nk \ |\ n\in\Bbb Z\right\} \cap \left\lbrace ...
8
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1answer
191 views

Surprising behavior of Leibniz formula for Pi (as Euler product)

I wrote a program to compute successive approximations of Pi using the following Euler product: π/4 = (3/4)*(5/4)*(7/8)*(11/12)*(13/12)... in which the ...
6
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2answers
2k views

Finding a point on Archimedean Spiral by its path length

I've been curious about Archimedean Spirals and their relations to Sacks Spirals and prime numbers. I would like to draw some visualizations of the points with a given distance from the center, ...
2
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1answer
59 views

connection between odd primes and a certain q-series

I posed a conjecture about odd primes and a certain q-serieshere.I thought it would be more appropriate ,if I could ask the converse of the aforementioned problem . Is $p$ an odd prime iff ...
1
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0answers
33 views

What is this quantity?

I wonder whether there is a closed form for $$-\sum_{k=1}^{\infty}\frac{\Delta^{k}\pi(x)}{k!}(-x)_k$$ where $\pi(x)$ is the prime-counting function and $(x)_k$ is the falling factorial. In other ...
3
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3answers
113 views

Is a prime to the power of a fraction always irrational?

Let $p$ be a prime number and let $x$ be a fraction, i.e. $x \in \mathbb{Q} \setminus \mathbb{N}$. It seems to be the case that $p^x$ is always irrational. How do I prove this?
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0answers
54 views

On $p^{\log_q n}$, where $p$ and $q$ are distinct primes

Let $p,q$ be distinct primes, $n>1$ an integer with $\log_q n $ irrational. It was, and probably still is, a conjecture that $p^{\log_q n}$ is non-integer. What progress has been made towards it?
7
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2answers
326 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 $\mathbb{N}$. ...
4
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1answer
128 views

conjecture about primes and a certain q-series.

Using wolfram Mathematica ,I observed an interesting and surprising property concerning prime numbers and q-series which I could not prove.Yet there is strong evidence supporting it. I would be happy ...
1
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0answers
61 views

Twin prime conjecture (Goldbach-Collatz remix)

Assuming Goldbach's conjecture, let's denote $r_{0}(n)$ for any integer $n$ greater than $1$ the smallest non negative integer $r$ such that both $n+r$ and $n-r$ are primes. Let $f$ be the map ...
0
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1answer
27 views

Does the smallest prime factor of a Fibonacci number appear in the Fibonacci sequence?

I thought of a way to tackle the problem of knowing whether there are infinitely many Fibonacci primes or not and this question came to my mind: does the smallest prime factor of any Fibonacci number ...
5
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2answers
368 views

Simple proof that $\pi$ is irrational - using prime factors of denominator

Simple proof that $\pi$ is irrational Consider the Gregory - Leibniz series for $\pi/4$: $$\frac \pi 4 = 1 - \frac 1 3 + \frac 1 5 + \cdots $$ Let $A_n/B_n$ be the irreducible fraction given by ...
0
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2answers
75 views

$x^4 + 4 y^4$ never a prime $>5$?

Let $x,y$ be nonzero integers. I could not find primes apart from $5$ of the form $x^4 + 4 y^4$. Why is that ? I know that if x and y are both not multiples of $5$ then it follows from fermat's ...
3
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2answers
41 views

If $p$ and $q$ are primes, which binomial coefficients $\binom{pq}{n}$ are divisible by $pq$?

If $p$ and $q$ are primes, which binomial coefficients $\binom{pq}{n}$, $1 \le n < pq$, are divisible by $pq$? In particular, if $p$ and $q$ are distinct odd primes, and $n$ is even, does $pq ...
6
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3answers
78 views

Arbitrarily large values for $|Li(x) - \pi(x)|$

I was wondering whether there are arbitrarily large values for the $|Li(x) - \pi(x)|$. I do know that $Li(x) - \pi(x)$ changes sign infinitely often, but this does not imply that the difference stays ...
35
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1answer
1k views

Are there any Mersenne primes, besides 3, that end in 3

It's clear that Mersenne primes can't end in $9$, since $2^n$ can't end in $0$, but $2^n$ can end in $4$ and $2^{n}-1$ would end in 3. From the list at http://mathworld.wolfram.com/MersennePrime.html ...
3
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1answer
84 views

Prove that $\mathbb{Q}[\sqrt{p}]$ is a field between $\mathbb{Q}$ and $\mathbb{R}$

I really have trouble understanding a task. We've got $p\in$ P, while P are all prime numbers. Now we construct a field $$\mathbb{Q}[\sqrt{p}]:=\{x+y\sqrt{p}:x,y \in \mathbb{Q}\}$$ The Task is to ...
0
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1answer
57 views

Find the natural numbers so that n=2*a^2 ,n=3*b^3 ,n=5*c^5.Number theory problem.

Well here it is i spend almost 3 hours on this one!! Find the general form of the natural numbers that are twice a square ,tripple of a cube and 5 times a 5-ith power.Who is the smaller of them?.What ...
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3answers
242 views

The number of possible factorizations of a positive integer.

Given a positive integer $n>1$ with prime factorization $$n=\prod_{p_i \text{ prime}}p_i^{k_i}, \space i\ge1, \space k_i \in \mathbb N^*$$ how can I compute the number of factorizations of ...
6
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1answer
154 views

Polynomial equations in $p$ and $q$ with $p,q$ primes

Is there a non zero polynomial $R \in \mathbb{Z}[X,Y]$ such that there exists an infinite number of pair $(p,q)$ with $p$ and $q$ primes, $p \neq q$ and $R(p,q)=0$ ? I know the curve must be of ...
5
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0answers
109 views

Approach to elliptic curve $y^2=x^3+1/4+p/a^2$

While taking a brute-force look at this question I discovered that it seems that almost every prime (I'll conjecture every prime larger than 20627) can be written as $p=w^2+wc+d$ for $w,c,d\in ...
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0answers
28 views

Sum over product of residues modulo two different bases

Let $M(x,y)$ be a modulo function. Specifically, $$ M(x,y) = \begin{cases} x, & \text{when } -\lfloor \frac y2\rfloor \leq x \lt \lfloor \frac y2\rfloor \\ M(x-y,y), & \text{when } x \geq ...
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2answers
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Is $\sqrt{m}$ irrational iff at least one prime occurs with an odd exponent in the factorisation of $m$?

Thinking about it, I think I found the following criterion for irrationality of $\sqrt{m}$ if $m$ is a positive integer. Let $p_1^{a_1}\cdots p_k^{a_k}$ be the prime factorization of $m$. Then ...
3
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1answer
57 views

Irrationality of the concatenation of decimal expansions 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 ...
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2answers
550 views

Prime Numbers and a Two-Player Game

In this question, $\mathbb{N}_0$ is the set of all nonnegative integers. The notation $\mathbb{N}$ is reserved for the set of all positive integers. Alex and Beth are playing the following game. ...
0
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1answer
43 views

$p^3 = 2009 + 47 * 2^q$ where p and q are primes

Solve the ecuations $p^3 = 2009 + 47 * 2^q$, where $p$ and $q$ are primes. Fermat's little theorem could help.
0
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1answer
42 views

Does $1+x+x^2 … x^{p -1}$ being a prime number imply p is prime? [duplicate]

Let $p$ and $x$ be two positive integers greater than 2. If it is given that the sum : $$1+x+x^2+x^3... x^{p -1}$$ is a prime, is it possible to prove or disprove that $p$ is prime? If so, what would ...
0
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2answers
64 views

Rational number that contains the sequence “$0123456789$”

Let $n$ be a rational number that contains the sequence "$0123456789$" in its decimal representation. Prove or disprove that there are $p,q \in \mathbb{N}$ such that $n = \frac{p}{q}$ and $q$ is a ...
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2answers
753 views

Is this formula for the $n^{th}$ prime number useful?

Is the below formula for the $n^{th}$ prime number in elementary functions useful somehow? $$p(n)=\sum _{a=2}^{2^n} \sin \left(\pi 2^{\left(n-\sum _{b=2}^a \frac{\sin ^2\left(\frac{\pi ...
2
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2answers
110 views

If $P$ and $Q$ are distinct primes, how to prove that $\sqrt{PQ}$ is irrational?

$P$ and $Q$ are two distinct prime numbers. How can I prove that $\sqrt{PQ}$ is an irrational number?
2
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1answer
51 views

Factorization of the semi-palprime $N = pq$

I define semi-palprime be a prime number that remains the prime when its digits are reversed, like $p = 13$, and its mate is $q = 31$. I know that number $N$, $ N ...
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2answers
83 views

Is $2^n-1$ always a prime for odd values of $n$? [duplicate]

Is $2^n-1$ always a prime for odd values of $n$? $n\not=1$ Taking some odd values of $n$, I observed outcome is coming as a prime number. How to verify it? Or at-least, is $2^n-1$ always coprime to ...
6
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1answer
60 views

$p+q\neq 1+pq$ for distinct odd primes $p$ and $q$

I'm trying to show that that $\sqrt p + \sqrt q$ cannot be written as a linear combination of $1$ and $\sqrt{pq}$ with rational coefficients, and I have boiled it down to showing that $p+q \neq 1+pq$ ...
7
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0answers
153 views

Where is The third Gisella prime?

A Gisella prime is a prime number obtained from concatenating the first $n$ natural numbers starting from $2$ and then replace each composite on that concatenation with the concatenation of its prime ...
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130 views

Primes of the form .$..55555444433322122333444455555…$

What is the smallest prime number of the form $...55555444433322122333444455555...$, where the concatenation runs through the first natural numbers, and where each decimal number $n$ being ...
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1answer
49 views

How one can solves an equation of the form: $ap_{n}+bn=c$

My question is: How can one solve an equation of the form: $$ap_{n}+bn=c$$ where $p_{n}$ is the $n^{th}$ prime number, $a,b$ and $c$ are integers.
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18answers
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Different ways to prove there are infinitely many primes?

This is just a curiosity. I have come across multiple proofs of the fact that there are infinitely many primes, some of them were quite trivial, but some others were really, really fancy. I'll show ...
3
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0answers
112 views

Richert’s theorem breaks down for $ n = 11 $

In 1949 H.-E.Richert proved (1) that every positive integer typeset structure is a sum of distinct primes. For more information please look at (2), and (3). However, if you consider $ n = 11 $, you ...
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1answer
67 views

Can the sum of 3 unique primes be expressed as the sum of 2 primes?

Let's consider the example, $$ 3 + 11 + 19 = 33 \\ 2 + 31 = 33 $$ we can see that there are cases where the sum of 3 primes be expressed as the sum of 2 primes. However, I couldn't find a case ...
9
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1answer
595 views

The largest possible prime gap?

What is the largest possible prime gap if we observe only 1000-digits numbers? There are few conjectures about this question but is there something that we can say and be absolutely sure of it?
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1answer
25 views

Theory behind prime generating function $p=an+b$, where $a, b$ are real coefficients

The Green–Tao theorem states that the sequence of prime numbers contains arbitrarily long arithmetic progressions. In other words there exist arithmetic progressions of primes, with k terms, where k ...
3
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2answers
37 views

Computational Complexity of Primality Checking

"PRIMES in P" proved that primality checking is in $P$. However, the CS 101 prime checking algorithm is to divide a number $n$ through all integers up to ${\sqrt n}$ , and if no results are whole ...