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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lattice walks with primes and composites

In the regular square lattice create a walk moving according the value of a counter. Consider two types of walks: In the first walk advance forward one unit if the counter is a composite number and ...
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Quadratic congruence prime numbers [on hold]

If $p$ is a prime number... a) show that $x^2 \equiv 1 \pmod{\!p}$ has only the following solutions: $x \equiv 1 \pmod{\!p}$ and $x \equiv -1 \pmod{\!p}$. b) show that $(p-1)! \equiv -1 ...
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Is $\pi(y)\pi(x+k)\ge\pi(x)\pi(y+k)$? [on hold]

Is it true that for some fixed $k\ge2$ and for all sufficiently large $x$ and $y$ with $y\ge x$ we have, $$\pi(y)\pi(x+k)\ge\pi(x)\pi(y+k)$$ where $\pi(x)$ is the prime counting function. I am ...
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Are these elementary assertions about quadratic intervals and prime factors true?

Assertion 1: Every quadratic interval is an even number from 2 to infinity. The interval size does not include the interval endpoints, $x^2$ and $(x+1)^2$. Therefore, every interval size is an even ...
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Counting the number of integers $x$ in a sequence of $30a$ consecutive integers where $\gcd(x(x+2),30)=1$ and $p \mid x(x+2)$ where $p \ge 7$

I was writing a computer program and I found that for all sequences that I tested the number of $x$ in a sequence of $30a$ consecutive integers for a prime $p$ is less than or equal to: ...
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Series with prime number second method

Today I made an exercise in which I had to prove that : $$\sum_{n \ge 1} \frac{1}{p_n} $$ where $p_n$ is a prime number ($p_1=2$) is divergent. Well it was really difficult and at the end of the ...
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Does this 'alternating' series with $\Lambda(n)$ converge for all $\Re(s)>0$?

The following equation is well known and valid for $\Re(s)>1$: $$\log\big(\zeta(s)\big)=\sum_{n=2}^\infty \dfrac{\Lambda(n)}{\log(n)\,n^s}$$ where $\Lambda(n)$ is the Von Mangoldt function. Take ...
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Prove that powers of any fixed prime $p$ contain arbitrarily many consecutive equal digits.

Prove that powers of any fixed prime $p$ contain arbitrarily many consecutive equal digits. It is an intuitive re-statement of Baltic Way 2012 (I think there are shortlists in Baltic Way every ...
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Geometric average of prime gaps?

The aritmetic average of prime gaps around $x$ is $\ln(x)$. What is the geometric average of prime gaps around $x$ ? Does that strongly depend on the conjectures about the smallest and largest gap ...
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How to prove that $a=z^{p}$ for some $z \in \mathbb{Z_{+}}$?

Claim : If for a positive, composite integer $a$ and an odd prime $p$, such that $\gcd(a,p)=1$, we are given $$ a^{p^{n-2}(p-1)} \equiv 1 \pmod {p^n} \ \forall \ n \geq 2 \ \ ;\ ...
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If $p$ is an odd prime with $(p - 1)/2$ primitive roots, is $p$ a Fermat prime?

If $p$ is an odd prime and there are $(p - 1)/2$ primitive roots modulo $p$, then is $p = 2^k + 1$ for some nonnegative integer $k$? This is the converse of a statement that I have already ...
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Do all prime numbers satisfy $p \mid (p-1)! + 1$? [duplicate]

If $m > 1$, $m \mid (m-1)! + 1$, then we can get the conclusion that $m$ is a prime number. But if we have a prime number $p$, can I get $p \mid (p-1)! + 1$? (I verify it when $p < 100000$, and ...
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Prime Numbers Primer [on hold]

This may not the appropriate site—but I thought Academia SE would be less appropriate. I'd like to begin working towards the ability to discover something novel about prime numbers. That is, I want ...
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How to efficiently list prime with a very specific property

I noticed that my phone number 06 xx xx xx xx is a prime number. Ok that cool ... But if you had the country code (+33 for france), ...
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odd prime numbers

For $m \geq 4$, set $P_m$ to be the set of all odd prime numbers strictly less than $m$ that do not divide $m$. For example, $P_4=\{3\}$, $P_7=\{3,5\}$, $P_{15}=\{7,11,13\}$. Now, for $n \geq 1$, ...
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Euler's Totient and divisors count function relationship when $[(\frac{\varphi(n)}{2}+1)\cdot(\frac{\tau(n)}{2}+1)] = n$

I am studying the Euler's totient function $\varphi(n)$ and the divisors count function, $\tau(n)$, also named $d(n)$, and recently opened a question (link here) about the following condition: ...
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Arithmetic progression - terms divisible by a prime.

If $p$ is a prime and $p \nmid b$, prove that in the arithmetic progression $a, a+b, a +2b, $ $a+3b, \ldots$, every $p^{th}$ term is divisible by $p$. I am given the hint that because ...
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The name given to the number 1 in the context of Primes and Composites

We give names to the sets of numbers called Primes and Composites. Is there a name for the number 1, in this context, seeing it is neither a Prime or Composite?
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Is there a ring - homomorphism $\mathbb{F}_p \rightarrow \mathbb{F}_q $ (p,q prime , $p \not= q$ )?

So we have two prime fields and seek a homomorphism between them. I assume that i have to find a homomorphism that is valid for all p,q prime , $p \not= q$, not just one for each choice. I would say ...
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Can I use integer frequencies in quadratic intervals to set a lower bound for primes? [closed]

I want to find out if the following arithmetic approach could produce a backdoor proof of Legendre’s Conjecture. There are two assumptions, Questions A and B, which are posed in the text and labeled ...
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Prime distribution around Riemann counting function

Is it true that the primes are normally distributed around the Riemann counting function $R(n)$ as a folded CDF? (Scaled $p_n-R(n)$)
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Proof that every quadratic residue has two roots, modulo a prime

Can someone provide a proof that every quadratic residue, when working in $\mathbb Z_p$, where $p$ is a prime, has exactly two roots? Indeed, there cannot be only one root as for any $a^2$, we know ...
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Number Theory Simple Proof

I am looking at a solution for a problem where the following line is stated but not explained, and I can not seem to make sense of it: If a prime $p\equiv 3\pmod 4$ then why is $\frac{p(p+1)}{2}$ ...
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Is this a meaningful approach to primes?

Motivation: I tend to be good at recognising patterns and I saw one with factorial: $$ n! = \prod_{i=1} p_i^{J(n,p_i)} $$ where $p_i$ is the $i$'th prime and $$ J(n,i)= \sum_{S=1}^\infty [n/i^S] $$ ...
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Finding the modulo of 801 [closed]

If $d_{k}(m)$ is the number of divisor of m that are congruent to $k$ modulo $4$. How can I find $d_{1}(801)$ and $d_{3}(801)$ .
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Highest ratio between consecutive prime numbers

Let r = p2/p1; where p1,p2 are consecutive prime numbers. What is the highest possible value of r? is there any consecutive prime numbers such that r > 5/3?
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Question about the Chinese Remainder Theorem and the residue class ring ${\bf Z}/p\# {\bf Z}$

In a question that I asked on MO, Terence Tao observed that: The Chinese Remainder Theorem tells us that the residue class ring ${\bf Z}/p\# {\bf Z}$ is isomorphic (as a ring) to the product of ...
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Aren't there obvious patterns in the primes that no one makes use of and what about this…

Let's take the sequence of naturals at or above two ($2, 3, 4, \dotsc$) and cross out just the primes $2$ and $3$, as well as all their multiples: $$\require{cancel}\cancel{2}, \cancel{3}, \cancel{4}, ...
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Find the smallest $N$ such that $\sum_{k=1}^N\frac{1}{p_k}>\pi$. (The $p_k$'s are the prime numbers.)

How to solve the following problem? Let $\{p_k\}_{k=1}^\infty$ be the set of primes (in increasing order). What is the smallest integer $N$ such that $$\sum_{k=1}^N \frac{1}{p_k}>\pi?$$ We ...
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Understanding the trivial primality test

I'm reading an algorithms book and I came across a code example for a primality test. The problem is that I couldn't understand the condition for the for-loop: ...
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Find all odd positive integers $n$ greater than $1$ such that for any coprime divisors …

Find all odd positive integers $n$ greater than $1$ such that for any coprime divisors $a$ and $b$ of $n$, the number $a + b − 1$ is also a divisor of $n$. This was taken from the Russian ...
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Question on Furstenberg topology on Z and P subspace of primes

Hi all I was given this question: I have Z (the integers) with the Furstenberg topology on it, i.e. the topology induced by non constant arithmetical progressions presented here, and I am asked to ...
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Contradicting $p|n $s.t $p > \sqrt n$

I have proved a basic theorem in prime numbers, If $n \ge 2$ and $n$ is composite, then it is divisible by some prime $p \le \sqrt n$. This is a fairly basic result, and then my textbook shows me how ...
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A problem about $e^{2\pi i \alpha_1}+e^{2\pi i \alpha_2}+\cdots+e^{2\pi i \alpha_N}=0$

Let $\alpha_i\in [0,1),\; i\in \{1,\cdots,N\}$ for some positive integer $N$, such that $$e^{2\pi i \alpha_1}+e^{2\pi i \alpha_2}+\cdots+e^{2\pi i \alpha_N}=0$$ and if for any non-empty proper subset ...
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Basic question about modular arithmetic applied to the divisor sum function $\sigma(n)$ when $n=5p$

While studying the divisor sum function $\sigma(n)$ (as the sum of the divisors of a number) I observed that the following expression seems to be true always (1): $\forall\ n=5p, p\in\Bbb P,\ p\gt ...
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Most Common Difference Between Two Consecutive Primes?

The question is as stated in the title. I was given this interesting problem by a friend of mine, but I don't know how to proceed with a solution. The immediate thought I had was that the most common ...
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If $n \in \mathbb N$, under what conditions are $n$ and $n+2$ relatively prime?

The Statement of the Problem: If $n \in \mathbb N$, under what conditions are $n$ and $n+2$ relatively prime? My Thoughts: I know that the answer is that $n$ must be odd. However, I'm not sure how ...
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Are there any known special properties of a number located between twin primes?

With the exception of $4$, every number located between twin primes is divisible by $6$. This one is obvious, but are there any other properties that can be ascribed to such numbers? A property may ...
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A miraculous number N

Of course we can talk about 1 digit prime numbers, 2 digit primes , 3 digit primes , and so on..., my question is : is there an N (N greater than zero) such that there are no N-digit prime numbers ? I ...
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Using Fermats prime numbers to prove that there is infinitely many prime numbers

A Fermat number $F_n$ is of the form $F_n = 2^{2^n} + 1$ Furthermore, $F_n = 2 + F_0F_1F_2......F_{n-1}$ Now I already proved that if $n \neq m$ then $\gcd(F_n,F_m) = 1$ Here is the proof Without ...
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Prove that if $p \mid a_1a_2 \ldots a_n$, then $p \mid a_j$ for some $j$ with $1 \le j \le n$

Let $p$ be a prime number and $a_1, a_2, \ldots, a_n$ be integers. Prove that if $p \mid a_1a_2 \ldots a_n$, then $p \mid a_j$ for some $j$ with $1 \leq j \leq n$. The hint was to use induction. ...
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Suppose $p$ is a prime number and $a$ is an integer. Show that if $p \mid a^n$, then $p^n \mid a^n$ for any $n \geq 1$?

I know that if $p \mid a^n$, I can say $a^n = pr$ for some integer $r$, you can also conclude that $\gcd(p, a^n) = p$, but I'm not sure how to use that information if I even can to show that $p^n \mid ...
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Prove that if $3^\frac{F_n-1}{2} \equiv -1 \pmod {F_n}$, then $F_n$ is prime [duplicate]

$F_n = 2^{2^n} + 1$ is a Fermat Number. Here is my attempt. We square each side of the congruences we get $$3^{F_n-1} \equiv 1 \pmod {F_n}$$ Now I already know that whenever $m \neq n$ then $ ...
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Do groups of prime order have one subgroup?

So let's say that I have a group of order $p$, where p is prime; does that group only have one subgroup? I've look at the wiki article and it says there's a trivial and actual solution, so can we ...
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the sum of the reciprocals of the primes

The sum of the reciprocals of the primes is $\sum \limits_{p}\frac{1}{p} \approx N \ln\ln(N)$ what about this sum where $p_{3}=3,p_{5}=5,p_{n}=\sum \limits^{N}_{j=5}\frac{1}{p_{j}} \sum ...
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Fermat's Little Theorem and prime divisors

Let $a,b\in\Bbb N$ and $a+b$ be an even number. Assume $a^2 - b^2 - a$ is an exact square, say $c^2$. Let $m = \frac {a+b}2$ and $n = \frac {a-b}2$. Then, $$(4m-1)(4n-1) = 4(4mn-m-n) + 1 = ...
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Prove that for all $n\in \mathbb N$ $(n>1)$ the number $n^4+4^n$ is not prime. [duplicate]

Prove that for all $n\in\mathbb N$ $(n>1)$ the number $n^4+4^n$ is not prime. Can someone give me some pointers?
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Squarefree products of a class of primes

Numbers which are the sum of two squares are the product of a square and a collection of distinct primes which are 1 or 2 mod 4. Landau proved that there are $\sim kx/\sqrt{\log x}$ such numbers up ...
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Prove that If $p$ is an odd prime, then any divisor of the Mersenne number $M_p = 2^p − 1$ is of the form $2kp + 1$

For example, $M_{11} = (2 · 1 · 11 + 1)(2 · 4 · 11 + 1)$ If $q$ is a prime divisor of $M_p$ then $\exists k \in Z$ such that $qk = 2^p-1$. Now $ord(2,q)$ is the smallest positive integer $a$ such ...
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Motivation for $r$ in AKS Primality Test

I've been reading up on the AKS primality test, and I understand the big ideas and proofs as they are pretty elementary number theory. I am confused about how to value of $r$ is selected. In the ...