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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1answer
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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 ...
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0answers
60 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 ...
2
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1answer
57 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 ...
0
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1answer
26 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 ...
0
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2answers
73 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 ...
1
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1answer
50 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, ...
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 ...
8
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1answer
189 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 ...
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 ...
4
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1answer
127 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 ...
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 ...
0
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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 ...
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
63 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 ...
5
votes
1answer
153 views

Conjectured compositeness tests for $N=k \cdot 2^n \pm c$

How to prove that these conjectures are true ? Definition : $\text{Let}~ P_m(x)=2^{-m}\cdot \left(\left(x-\sqrt{x^2-4}\right)^{m}+\left(x+\sqrt{x^2-4}\right)^{m}\right)~ , \text{where}~ m ...
4
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3answers
84 views

Prove $\log_7 n$ is either an integer or irrational

I have been trying to prove a certain claim and have hit a wall. Here is the claim... Claim: If $n$ is a positive integer then $\log_{7}n$ is an integer or it is irrational Proof ...
1
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2answers
80 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 ...
2
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1answer
46 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 ...
5
votes
1answer
57 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
149 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 ...
4
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0answers
127 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 ...
1
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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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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 ...
1
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1answer
65 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 ...
1
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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 ...
3
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1answer
35 views

Compute $\lim_{N\to \infty}N^2/\left(\sum_{\text{primes }\leq p_N}p\right)$, where $p_N$ is the $N$th prime number, and another related limit

It is well known that, where $p_k$ is the $k$th prime number (this is $2 = p_1 < p_2 < p_3 < \cdots$), the following Proposition. The series of reciprocals of primes ...
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0answers
127 views

Conjectured compositeness tests for $N=b^n \pm b \pm 1$

How to prove that these conjectures are true ? Definition : $\text{Let}~ P_m(x)=2^{-m}\cdot \left(\left(x-\sqrt{x^2-4}\right)^{m}+\left(x+\sqrt{x^2-4}\right)^{m}\right)~ , \text{where}~ m ...
1
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1answer
31 views

Strengthening or consequence of Goldbach?

Is the conjecture that every integer $n \equiv 2 \bmod{4}$ greater than $6$ can be expressed as the sum of two primes of the form $4n+3$ a strengthening, or a consequence of Goldbach?
2
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1answer
39 views

Longest sequence of primes with no difference greater than $2n$

Let $N\ge 1$ be a natural number. The object is to find the longest possible sequence of prime numbers $p_1<p_2<...<p_n$ such that $p_{i+1}-p_i\leq 2N$ for $i=1,...,n-1$. In other words, ...
0
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1answer
57 views

Improved Betrand's postulate

I want to show that $2p_{n-2} \geq p_{n}-1$... Bertand's postulate shows us that $4p_{n-2}\geq p_{n}$ but can we improve on this? any ideas?
5
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1answer
205 views

Wilson's Theorem - Why only for primes? [closed]

Why is it true that Wilson's Theorem only holds for prime numbers? I read a proof of it, and it did not seem to cater to that aspect of the theorem.
0
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1answer
31 views

Prime counting function [duplicate]

How much of an impact would the discovery of an exact formula that is equivalent to the prime counting function have on the mathematics community and acedemia as a whole?
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4answers
1k views

What is the symbol for primes?

Although there isn't much difference between $\mathbb{Z},\mathbb{N},\mathbb{I}$, they are well-known, and each one gets its own distinguished symbol. Is there any reason that primes don't get their ...
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0answers
40 views

Puzzle on multiplying by fixed values to reach a target number.

So, this one's tricky. There's a keycode combination, and there are six buttons. Each button multiplies the base number of 1 by their respective multipliers (see below). Once the result number gets ...
10
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2answers
220 views

Proof of infinitude of prime elements?

All proofs of infinitude of primes which I know of essentially prove that there are infinitely many irreducible elements of $\Bbb Z$, and with this goal in mind we can very easily extend this proof to ...
9
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1answer
153 views

What do we know about the first occurrences of prime gaps?

Are there any conjectures from which we can infer something about the first occurrences of prime gaps length $n$ and their distribution? I've made an interesting graph of these values to make this ...
0
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1answer
37 views

question about first occurring prime gaps

If a prime gap $g(p)$ is the first occurring prime gap of it's size, does this imply that it is also the largest gap below $p$? In other words, is the set of first occurring prime gaps contained ...
5
votes
2answers
87 views

Why is it that the product of first N prime numbers + 1 another prime? [duplicate]

Recently I came across this proof for fact that primes are infinite. It's a proof by contradiction. The proof assumes that primes are finite and there is a prime M which is larger than any prime out ...
3
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2answers
67 views

Indexes of prime Fibonacci numbers

I found this on Mathworld, but I can't seem to find any proof, either on StackExchange, nor any other site: Why do all Fibonacci primes, except for $F_4=3$, have prime indexes (with $F_0=0$)? My ...
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1answer
32 views

Fermat primality test and Fermat pseudoprime

What is the difference between Fermat primality test and Fermat pseudoprime?Can anyone explain me how we use them ?
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1answer
45 views

Prime Number Algorithm

function isPrime(n) { // If n is less than 2 or not an integer then by definition cannot be prime. if (n < 2) {return false} if (n != Math.round(n)) {return false} // Now assume that n is ...
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0answers
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Calculating $n$-th $q:P(q)=p \in \Bbb P$

Let $P(x)$ denote the number of ways of writing an integer $x$ as a sum of positive integers (where permutation of the array of integers in the sum doesn't count). Ex: $P(1)=1, P(2)=2,P(4)=5$. Let ...
1
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2answers
85 views

The product of two prime numbers

I have two expressions (both of which have a term raised to the power of $n$) and I am trying to prove that they can't be prime numbers at the same time for $n>2$. I can't post the expressions, ...
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2answers
31 views

Is there a method to determine a prime number containing the first n digits?

For example, the number $10243$ is prime and contains the digits '0,' '1,' '2,' '3,' and '4.' Similarly, the number $20143$ is prime. Is there a method to determine whether a prime number exists ...
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3answers
30 views

Finding all the divisors of $a$ by decomposing it into the product $p^{\alpha_1}_{1} \cdot p^{\alpha_2}_{2} \cdots p^{\alpha_r}_{r}$

I'm trying to prove the following statement regarding the fundamental facts of prime numbers, but I don't really understand the relationship between $a$ and $b$. In order to find all the divisors ...
4
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0answers
63 views

What is the next prime with this form? [duplicate]

The following are primes, $$P_1 = 2^2 + 3^3$$ $$P_2 = 2^2 + 3^3 +5^5 + 7^7$$ After these two, the only prime of such form I've found is, $$P_3 = 2^2 + 3^3 + 5^5 + 7^7 + 11^{11} + 13^{13} +\dots+ ...
10
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2answers
548 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. ...
5
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2answers
251 views

Inequality involving a product over the primes

Is there someone who is able to prove the following statement? $$\prod_{m=1}^n \dfrac{p_m-1}{p_m} \leq \dfrac{1}{\ln(n)}$$ for all integers $n >1$ where $p_m$ is the $m$-th prime number.