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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Proving Goldbach's conjecture (hypothetically) - probabilistic argument

Part $1$. If $\pi(n) \sim \frac{n}{\ln(n)}$ by the prime number theorem, can we treat $\frac{1}{\ln(n)} $ as the probability that a number less than $n$ is a prime number? Say we have some operation ...
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Minimum amount of primes between squared primes

Conjecture: $\forall $ $p_{n}$, $p_{n+1} \in \mathbb{P}$, $\:$ $p^2_{n+1} = p^2_{n} +\omega_{n} p_{n} + \phi_{n} : \phi_{n} , \omega_{n} \in \mathbb{N} $ and $ \phi_{n} < p_{n}$, $\:$ ...
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241 views

Density of primes in intervals less than primorial numbers

Looking at the interval of the natural numbers $ [1, p_{n}$#$] $; $\frac{1}{2}$ of the elements of this set will be even, and $\frac{1}{2}$ will be odd. $\frac{1}{3}$ of the elements of this set will ...
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198 views

Prime gaps with respect to the squared primes

Conjecture If we have two consecutive prime numbers $p_{a}$ and $p_{a+1}$, and two other consecutive primes $p_n$ and $p_{n+1}$, so that $p_{a} < p_{a+1} < p^2_{n+1}$, then $p_{a+1} - ...
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Existence of a prime between $ap$ and $(a+1)p$ - generalization of Bertrand's postulate

Conjecture: There exists at least one prime number $p_{m}$ : $ap_{n} < p_{m} < (a+1)p_{n}$, $\forall$ $a \in \mathbb{N}$ and $\forall$ $p_{n}$ $\in \mathbb{P} $ if $(a+1)p_{n} < ...
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184 views

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

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.
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Intersection between the sums of the first 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 ...
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29 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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Goldbach conjecture related thinking

Background The sequences $a_k = 2(p-1)k + p$ yield many prime numbers when $p$ is a prime number. (Test it) $2p \equiv 2 \bmod{(2p-2)} \equiv 2 \bmod {(2(p-1))}$ Direction Following from ...
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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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Is this statement equivalent to Goldbach's conjecture

Given a number $n\ge 3$, then one of these is true: \begin{equation} \begin{cases}2n = (6m-1)+P, \ \ \ P \in \mathbb P, \ 6m-1 \in \mathbb P, \ 6m+1 \in \mathbb P \ \ \ \ (1) \\ 2n-1 \in \mathbb P, \ ...
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Question on occurrences of prime gaps [on hold]

Why is the number of times a prime gap $p_{n} - p_{n-1}$ is above $\ln(p_{n-1})$ always the same as the number of times it occurs below $\ln(p_{n-1})$?
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+100

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. ...
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Sheldon Cooper Primes

On the $73^{\text{rd}}$ episode of the Big Bang Theory, Dr. Sheldon Cooper, an astrophysicist portrayed by Jim Parsons $(1973 - \stackrel{\text{hopefully}}{2073})$ revealed his favorite number to be ...
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Does the set of $m \in Max(ord_n(k))$ for every $n$ without primitive roots contain a pair of primes $p_1+p_2=n$?

I have made the following observation: for those n even numbers that do not have primitive roots modulo n ,$Pr(n)$, the set $M(n)$ of those $k$ having a maximum multiplicative order $ord_n(k)$ ...
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Why the Riemann hypothesis doesn't imply Goldbach?

I'm interested in number theory, and everyone seems to be saying that "It's all about the Riemann hypothesis (RH)". I started to agree with this, but my question is: Why then doesn't RH imply the ...
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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 ...
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1answer
186 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 ...
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1answer
138 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 ...
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2answers
732 views

Infinitely many primes of the form $8n+1$

I'm looking at this funny little problem involving proving the existence of an infinite number of primes of a certain form: Prove that there are infinitely many prime numbers expressible in the ...
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Summation of prime multiples less than n

How can I sum the following $$ \sum (2^x\cdot3^y\cdot5^z\cdot7^w\cdot\prod_1^m(p_i^{a_i})) $$ with $$2^x\cdot3^y\cdot5^z\cdot7^w\cdot\prod_1^m(p_i^{a_i}) \le n$$ where $p_i \ge 11$ are list of fixed ...
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Test about prime gaps: which conclusions can be drawn from the results?

I did the following test: For every prime, take the prime gap distance $dp$ to the previous prime and the next prime $dn$, then calculate $a=(pp\ mod\ dp)$ and $b=(np\ mod\ dn)$. If $a$ or $b$ ...
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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 ...
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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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2answers
84 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 ...
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1answer
28 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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Is $1992! - 1$ prime?

Consider the factorials, defined inductively by $1! = 0! = 1$ and $n! = n\cdot(n-1)!$ for $n \geq 2$. Question: Is $1992!-1$ a prime number? The question is from a book, maybe is contest math ...
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70 views

Sets with $n$ prime numbers [on hold]

Let $n>2$ a natural number. We define the following sets: $$S=\{1 \leq a \leq n : (a,n)=1, a^{n-1} \not\equiv 1\pmod n\} \\ T=\{1 \leq b \leq n : (b,n)=1, b^{n-1} \equiv 1 \pmod n\}$$ ...
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Why does symmetry happen in reset-based random walks?

Studying the basic concepts about random walks / brownian motion, and based on the idea of a Möbius-based walk in Wolfram's website, I wanted to try my own version of it in Python to compare it with ...
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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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Are primes randomly distributed?

There is a famous citation that says "It is evident that the primes are randomly distributed but, unfortunately, we don't know what 'random' means." R. C. Vaughan (February 1990) I have this very ...
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Is there a definition about prime pairs $(p,q)$,$\quad q=(n \quad mod \quad p)$ and $2p+q=n (odd)$?

I am studying congruences and I have observed this kind of prime pairs $(p,q)$ related to odd numbers. Here is the definition: $\forall n=1+2k,\quad k\ge5,\quad k \in \mathbb N, \quad \exists$ $p ...
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1answer
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Is the Green-Tao theorem valid for arithmetic progressions of numbers whose Möbius value $\mu(n)=-1$?

I am reading the basic concepts of the Green-Tao theorem (and also reading the previous questions at MSE about the corollaries of the theorem). According to the Wikipedia, the theorem can be stated ...
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Show that $n!+1$ has a prime factor $\;>n$; showthat there are infinitely many primes

I don't know how to prove this and it's really bugging me. Thanks to anybody that can help! Let $n$ be any natural number. Prove that $n! + 1$ contains a prime factor greater than $n$ and use that to ...
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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.
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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 ...
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Can a number be equal to the sum of the squares of its prime divisors?

If $$n=p_1^{a_1}\cdots p_k^{a_k},$$ then define $$f(n):=p_1^2+\cdots+p_k^2$$ So, $f(n)$ is the sum of the squares of the prime divisors of $n$. For which natural numbers $n\ge 2$ do we have ...
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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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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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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 ...
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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+ ...
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Is $a=\frac{1992!-1}{3449\times 8627}$ a prime number?

Is $a=\dfrac{1992!-1}{3449\times 8627}$ a prime number ? This is a natural follow-up to that recent MSE question We know that $a$ has $5702$ digits and no prime divisor $<10^6$.
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Is this a valid equivalent expression of the twin prime conjecture?

The twin prime conjecture states basically that it is possible to find two primes $p$, $p+2$ at a distance $2$ that are as big as wanted (Wikipedia). I am learning about the basic properties ...
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What is this pattern in this calculation using primes?

So I was bored and when I get bored I write small programs that calculate something. This time I did this: I searched for the amount of primes bellow 10,000,000 using Sieve of Eratosthenes starting ...
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Efficient way to compute $\sum_{i=1}^n \varphi(i) $

Given some upper bound $n$ is there an efficient way to calculate the following: $$\sum_{i=1}^n \varphi(i) $$ I am aware that: $$\sum_{i=1}^n \varphi(i) = \frac 12 \left( 1+\sum_{i=1}^n \mu(i) ...
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what is Prime Gaps relationship with number 6?

Out of the 78499 prime number under 1 million. There are 32821 prime gaps (difference between two consecutive prime numbers) of a multiple 6. A bar chart of differences and frequency of occurrence ...
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How to determine if a number $A$ is divisible by all the prime factors of $B$?

How to determine if a number $A$ is divisible by all the prime factors of $B$? For example: $120,75$ $A=120=2^3\times3\times5$ and $B=75=3\times5^2$ Therefore yes, $A$ is divisible by the prime ...
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inequality on Gaussian prime

Consider any Gaussian prime $p$ (except $|p|=\sqrt{2}$). If we have $|x|\leq|p|+0.5$, where $x$ is a nonzero Gaussian integer, can we prove $|x|\leq|p|$?
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Is there an advantage in using continued fractions for Catalan or Fibonacci-Lucas primality tests?

I am studying the basic theory about continued fractions and also reviewed here at MSE former questions to learn more. While reviewing the questions and answers, I found references to the Fibonacci ...