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

1
vote
3answers
32 views

First index of number in that arithmetic progression which is a multiple of the given prime number

I have a prime number $p$, an arithmetic progression starting at $a$ with common difference $d$. How to find the first index of a term in that arithmetic progression which is a multiple of the given ...
0
votes
1answer
30 views

Probability and Prime Numbers

What is the probability that a positive divisor of 8748 million is the product of exactly 20 non-distinct primes?
0
votes
0answers
20 views

A Question Related to Zsigmondy's Theorem

I am wondering if there is a way to prove the following statement, which bares some resemblance to Zsigmondy's Theorem. I am not sure if the statement is true, but it seems as though it should be. ...
3
votes
0answers
39 views

Euler's proof of divergence of sum of reciprocals of primes

On Wikipedia at link currently is: \begin{align} \ln \left( \sum_{n=1}^\infty \frac{1}{n}\right) & {} = \ln\left( \prod_p \frac{1}{1-p^{-1}}\right) = -\sum_p \ln \left( 1-\frac{1}{p}\right) \\ ...
2
votes
0answers
24 views

Are there any primes that are never a factor of a Carmichael number?

Is there a prime number $p$ that $p > 2$, and in which $p$ is a never a factor of any Carmichael number $C_n$: (p ∤ $C_n$) Extended this to all numbers $m$, instead of just $p$, will prove the ...
0
votes
1answer
74 views

Number theory proof regarding primes and the number of digits of the prime [duplicate]

How would you prove that if given a prime each of whose (decimal) digits is equal to $1$, then the number of its digits is a prime. (It is not known if there exists infinitely many such prime)
0
votes
1answer
138 views

Let $x = 2441921$. Factor $x$ into a product of primes.

Let $x = 2441921$. Factor $x$ into a product of primes. I found that: $1519^2 −x=−134560= −2^5 ·5 · 29^2$ and $1541^2 −x=−67240= −2^3 · 5 · 41^2$. I am trying to figure out what to do from here. ...
0
votes
2answers
57 views

Represent a prime number $p$ congruent to $1$ $\pmod{3}$ by a sum of a square and $3$ times a square

I want to have a proof of the fact that each prime number is the sum of a square and three times a square (Euler). Context I read the answer to my former question about the number of points on an ...
1
vote
4answers
69 views

$p$ prime, $p\mid a^k \Rightarrow p^k\mid a^k$

Suppose $p$ is a prime and $a$ and $k$ are positive integers. Prove that if $pa^k$ then $p^k\mid a^k$. I have already proven that if $a,b,n\in\mathbb{N}$ and if $a^n\mid b^n$, then $a\mid b$. I tried ...
4
votes
0answers
40 views

Computing question: A quadratic which gives primes [on hold]

This is about Project Euler Problem 27. The question is: Considering quadratics of the form $n^2 + an + b$, where $\lvert a \rvert < 1000$ and $\lvert b \rvert < 1000$ Find the product ...
1
vote
1answer
21 views

Length of smallest repunits divisible by primes

I want to prove this statement from Wikipedia: It was found very early on that for any prime p greater than 5, the period of the decimal expansion of 1/p is equal to the length of the smallest ...
3
votes
2answers
52 views

Prove that every non-prime natural number $ > 1$ can be written in the form of $n+(n+2)+(n+4)+…+(n+2m) = p$

I'm trying to prove that every non-prime natural number greater than $1$ can is equal to a sum of consecutive even or odd numbers. This can be resumed in : « $p,m,n \in ℕ$» , «$p > 1$» , «$n > ...
3
votes
0answers
29 views

Probability and sums of prime factors

Of the first N natural numbers, we select two different numbers at random. We'll call the greater one A and the lesser one B. What is the probability (P) that the sum of A's prime factors is LESS than ...
0
votes
0answers
29 views

Find a criterion for the primes p such that (5/p) = 1. [duplicate]

Find a criterion for the primes p such that (5/p) = 1. I don't understand this question it is like Determine all primes P such that (5/p)=1 I appreciate any help
-2
votes
1answer
102 views

A Twin Primes Sequence [on hold]

How to prove the relation below and is it enough significant to prove the infinitude of the twin primes? For every twin primes $x,y$ there exists $\alpha,\beta$ positive integers such that ...
12
votes
0answers
142 views
+150

Can a Mersenne number ever be a Carmichael number?

Can a Mersenne number ever be a Carmichael number? More specifically, can a composite number $m$ of the form $2^n-1$ ever pass the test: $a^{m-1} \equiv 1 \mod m$ for all intergers $a >1$ ...
28
votes
2answers
4k views

Are weird numbers more rare than prime numbers?

By taking a look at the first few weird numbers: $$(70, 836, 4030, 5830, 7192, 7912, 9272, 10430)$$ It is certain that prime numbers occurs more often within this range of numbers. But are weird ...
4
votes
5answers
125 views

What is the proof to the fact that all prime numbers are 1 above or below a 6 multiple? [duplicate]

I was just having an argument with my friend and I dunno how we got here. But he suddenly said all primes are 1 above or below a multiple of 6. At first I tried a lot of primes but couldn't disprove ...
0
votes
1answer
25 views

Divisibility of $a^p-r$ and $a^q-r$ by the primes $p,q$

Let $p, q$ be prime and $a$ some positive integer such that $a = pq + r$ where $r$ is the remainder. Show that $p \mid a^p – r$ and $q \mid a^q – r$. Example: $p = 3$ and $q = 5$, $a = 17$ and $r = ...
-1
votes
0answers
64 views

Andrica's conjecture implies Bertrand's Postulate?

Let $p_n$ denote the $n$th prime. Recall Andrica's conjecture, which states that $$\sqrt{p_{n+1}}-\sqrt{p_n}<1\quad\text{ for all }\,n.$$ I think Andrica's conjecture implies Bertrand's postulate. ...
3
votes
1answer
50 views

If $p$ is prime, prove that $\exists k\in\lbrace 5,-7,9,-11,..\rbrace$ in $(\mathbb{Z}/p\mathbb{Z})^*$ so that the Legendre symbol $(\frac{k}{p})=-1$

The BSPW primality test, when given $p$ as input, iterates over $k \in \lbrace 5,-7,9,-11,...\rbrace$ as long as the Legendre symbol $(\frac{k}{p})=1$. If $(\frac{k}{p})=0$, it returns "composite". So ...
-3
votes
1answer
82 views

A new Set of Primes [on hold]

Considering the following do there exists a fatal error? There exists infinitely many primes $\zeta$ and $\beta$ such that ...
-3
votes
1answer
49 views

Billion-Digit Prime Number [on hold]

I am trying to identify a billion-digit prime number in the form 10^(10^9) + x. This number could be found in the form NextPrime[10^(10^9)], but I do not want to look at a billion digits. To do this, ...
3
votes
0answers
89 views

The n-envelope problem

This is original problem: You have n number of envelopes, and 100 $1 bills. you have to put these bills in the envelopes in such a way that any amount between 1 to 100 can be reached just by ...
4
votes
0answers
67 views

Are large prime numbers kept secret? [duplicate]

I've read several times that modern cryptography is based on the fact that multiplying two primes is easy, whereas getting the prime factorization of a random big integer is very hard. (see here) ...
1
vote
1answer
53 views

Limit of Sum $\sum \frac{p(k+1) - p(k)}{p(k)^2}$ ? (corrected)

This is a re-post with my (corrected) earlier typo in red as a reminder. The question is about $$(A)\hspace{10mm}\lim_{n \to \infty} \sum_{k=\color{red}2}^{n} \frac{p(k +1) - p(k)}{p(k)^2} = ...
3
votes
1answer
104 views

Showing that if $p$ is prime, then $(p^4 + 4)$ can't be prime

I want to show that if $p$ is prime, then $(p^4 + 4)$ can't be prime. I guess Fermat's little theorem may help, but I can't figure out how to use it for the proof. Can anyone point me in the right ...
2
votes
1answer
96 views

Are there infinitely many prime numbers? [duplicate]

I will post my own answer below. That should not deter others from answering. There are many ways to prove this.
1
vote
1answer
26 views

Sizes of Blocks of Consecutive Integers Divisible by at Least One Prime Less than or Equal to $r$.

Let $f(r)$ be the largest integer such that there exists a block of $f(r)$ consecutive integers each divisible by some prime that is less than or equal to $r$. For example, $f(2)=1$ because it is ...
3
votes
0answers
53 views

An Inequality with Prime Numbers

Let $m\geq 8$ be an integer, and let $q$ be the smallest prime that is greater than $m$. Let $p$ be a prime that satisfies $q\leq p<q^2$, and let $p_0$ be the smallest prime such that ...
2
votes
1answer
42 views

For an integer $n \geq 2$, let $m$ be the largest positive integer less than $n$ such that $m \mid n$. [closed]

Consider an integer $n \geq 2$, and $m$ the largest positive integer less than $n$ such that $m \mid n$. Then $n = mk$ for some positive integer $k$. Prove that $k$ is a prime number.
1
vote
2answers
216 views

Visualization of Eratosthenes’ sieve

In otherwise great paper on prime numbers, I found following visualization of Eratosthenes’ sieve: I found it somewhat scary and confusing. Is there any better visualization of Eratosthenes’ sieve ...
1
vote
0answers
43 views

Prime bounds under RH

Continuing from here, since $$ \sum_{k=1}^{\lfloor\log(n)\rfloor}\dfrac{\pi(n^{1/k})}{k}=\operatorname{li}(n)-\sum_{k=1}^{\infty}2\ ...
0
votes
1answer
37 views

Existence of at least one prime for all sequences in the family of sequences

Prove or disprove that for a fixed $n \in N$, there exists at least one prime among the integers of the form $2^{k}n+2^k-1$ for an arbitrary $k \in N$.
0
votes
2answers
126 views

$9n^2-4$ only generates one prime? Why?

Instead of doing the work I was supposed to be doing, I played around with some numbers, and I noticed that for $n\in\mathbb{N}$, $9n^2-4$ only seems to generate a prime for $n=1$. Can anyone ...
0
votes
4answers
111 views

Is at least one of $6k + 1$ or $6k-1$ prime?

We know that any prime number ( $> 2,3$) can be written in the form $6k+1$ or $6k-1$. Is it necessary that at least one of $6k+1$ or $6k-1$ is a prime number ?
0
votes
1answer
39 views

Q: Understanding Answer of 2012 AMC 8 - #18

The problem is: "What is the smallest positive integer that is neither prime nor square and that has no prime factor less than 50?". The solution for this problem goes like this: "Since the integer ...
0
votes
2answers
109 views

Congruence modulo primes or in a polynomial ring over ${\rm GF}(2)$

Let $p, q$ be primes. Then the linear congruence $$ap \equiv r\pmod q$$ can be solved for $a\in\mathbb Z$ and will have a unique solution for each value of $r$ such that $0\leqslant a<q$. Am I ...
1
vote
2answers
31 views

Prove that 1 is the only “common” divisor of the integers n and n+2

Let n be any odd integer. Prove that 1 is the only "common" divisor of the integers n and n+2. I think you have to find gcd(n, n+2) and say that since n odd then then n+2 will also be odd. Thus n + ...
0
votes
0answers
44 views

Fast check of safe primes or Sophie Germain primes

If $p=2q +1$ with $p,q$ prime then $p$ is called safe prime and $q$ is a Sophie Germain prime. I want a faster algorithm for a safe prime test than doing two primality checks for $p$ and $q$. In ...
7
votes
1answer
165 views

Prove $18080108080 \sum_{k=0}^{1560-1} 10^{10k}+1$ is prime

I saw this fact on twitter: I would like to know how one would show this number is prime. Is there an elementary way to show that this number is prime? Is there a simplified primality testing ...
2
votes
1answer
82 views

Prime number upper bound

I am reading some written notes about a proof I do not understand, maybe some informations are missing. The result that has to be proved is the following: if $p_n$ is the $n$-th prime number, ...
1
vote
0answers
83 views

How I could transform this into product over primes :$s_p$= $\frac{1}{2^2-1}+\frac{1}{2^3-1}+…\frac{1}{2^p-1}$?

1)Can I transforme this sum into product OVER primes:$s_p$= $\frac{1}{2^2-1}+\frac{1}{2^3-1}+....\frac{1}{2^p-1}$ ? Note : p is prime number and ${2^p-1}$ is prime 2)I would be interest to know ...
0
votes
1answer
27 views

Largest $k$ such that $(p-k) = \lceil \sqrt{(p-2k) p} \rceil$

Assume $p \in \mathbb P.$ Assume $0<p-2k<p$ and the next square larger than $p(p-2k)$ is $(p-k)^2$. It is trivial to show that $p(p-2k)+k^2$ is a square. Simply $p(p-2k)+k^2 = (p-k)^2.$ ...
7
votes
2answers
259 views

Is $7^{8}+8^{9}+9^{7}+1$ a prime? (no computer usage allowed)

Prove or disprove that $$7^{8}+8^{9}+9^{7}+1$$ is a prime number, without using a computer. I tried to transform $n^{n+1}+(n+1)^{n+2}+(n+2)^{n}+1$, unsuccessfully, no useful conclusion.
1
vote
2answers
40 views

The $i^{th}$ prime in a given ring R

When I say that $p_1=2$, I mean that the first prime in the standard ring of integers $(\mathbb{Z},*,+)$ is $2$. I was wondering whether the notion of ordering the primes like this can be generalized ...
5
votes
2answers
116 views

Is this Goldbach-type problem easy to solve?

Problem: Given an odd prime number $p$, are there odd prime numbers $q$, $p'$, $q'$ such that $\{p,q\} \neq \{ p',q'\}$ and $p+q = p'+q'$ ? This comment informs that it's an obvious ...
3
votes
2answers
99 views

Given $N$, what is the next prime $p$?

Certain data structures in programming related to collections operate in an optimal way if they have prime number of elements. This means if a program (programmer) requires $N$ (any natural number) ...
5
votes
1answer
317 views

Identity for frequency of integers with smallest prime(n) divisor

An identity for A038110 and A038111: $$ \frac{\phi(e^{\psi(p_{n}-1)})}{e^{\psi(p_{n})}}=\frac{\prod _k^{n-1} \left(1-\frac{1}{p_k}\right)}{p_n}, $$ where $\psi(\cdot)$ is the second Chebyshev function ...
4
votes
1answer
202 views

Is $2013^{2014}+2014^{2015}+2015^{2013}+1$ a prime? (usage of a computer not allowed)

Prove or disprove: $$2013^{2014}+2014^{2015}+2015^{2013}+1$$ is a prime number, without using a computer. I tried to transform the expression $n^{n+1}+(n+1)^{n+2}+(n+2)^{n}+1$, but couldn't reach ...