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Questions tagged [prime-numbers]

Prime numbers are natural numbers greater than 1 not divisible by any smaller number other than 1. This tag is intended for questions about, related to, or involving prime numbers.

A prime number (or a prime) is an element of the greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number ... The fundamental theorem of arithmetic establishes the central role of primes in :

Any integer greater than 1 can be expressed as a product of primes that is unique up to ordering.

Here you get the first 50 millions of primes.

The concept of prime numbers is extended in ring theory, where an element $p$ of a ring $R$ is prime if and only if whenever $p\mid ab$, then $p\mid a$ or $p\mid b$.

One can easily see that this extends the definition of prime numbers in the natural numbers.

12562 questions
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About $x(\ln(x\ln(x))-1)5$ and better results

I need some tips about this: It has been proved that (1) $$x(\ln(x\ln(x))-1)5$$ Is there a better results? Thanks!
Andrea
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Why $\sum_{k=1}^\infty \left\lfloor\frac{n}{p^k}\right\rfloor≤\frac{n}{p-1}$?

I need some help: can someone tell me why $$\sum_{k=1}^\infty \left\lfloor\frac{n}{p^k}\right\rfloor≤\frac{n}{p-1}$$ I found this inequality in Wikipedia, and I want to know if it's true, thanks!
Andrea
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primes congruent to 5 mod 6 and 1 mod p

I believe that for every prime $p\geq 5$ there exists at least one prime $q$ that is both congruent to 1 mod $p$ and congruent to 5 mod 6. It's well known that there are an infinite number of primes congruent to 1 mod $p$ and there are an infinite…
Ian Parberry
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Find one possible pair of values for x and y. If x,y and x-y are two-digit numbers. x is a square number, y is a cbe number and x-y is a prime number

Find one possible pair of values for x and y. If x,y and x-y are two-digit numbers. x is a square number, y is a cube number and x-y is a prime number. Is it as easy as I am thinking it is? Or I am trying to complicate things by looking for the…
laila
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Approximating prime number function

What is the best way to approximate how many primes there are less than $2^{43112609}-1$? I know that one can use prime number theorem. I also found that in the Internet that $\pi (10^{24})=18435599767349200867866$ and then one can use Loo's theorem…
student
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prove that $p^2-1$ is divisible by $24$ if $p$ is a prime greater than $3$

How to prove that $p^2-1$ is divisible by $24$ if $p$ is a prime number greater than $3$?
tim_yng
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Sum of certain two-digit primes with prime digits

Let $P$ be a two-digit prime number less than $100$ such that both digits are prime numbers. What is the sum of all such numbers, $P$? Is there a quick way to solve this problem without listing all the numbers?
Hector
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Infinite Prime Numbers: With Fermat Numbers

Suppose that the Fermat numbers $F_m$ are pairwise relatively prime. Can someone help me prove, given this, that there are infinitely many primes.
jento1
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Fermat Numbers are pairwise coprime $\implies$ infinitely many primes

Given that the Fermat numbers $F_m$ are pairwise relatively prime. Prove that there are infinitely many primes.
Alexis Dailey
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Question about Euclid's infinite prime proof

Suppose that $p_1=2 < p_2 = 3 < \cdots < p_r$ are all of the primes. Let $P = p_1p_2...p_r+1$ and let $p_s$ be a prime dividing $P$ where $p_s$ is not in our original list $p_1, p_2, \cdots, p_r$. If $p_s$ can divide $p$, wouldn't it be able to…
Don Larynx
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Prime number minus 1 is an even number?

Is it true that for every prime number $p$ (except $p = 2$), that $p-1$ is an even number? I tried it in R (code below) for the first 168 primes (found on wikipedia) and it seems to hold, but I'm not sure if it is always true (maybe its a naturally…
neuteich
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Method to solving this proof with a java app

I'm writing a program to solve this proof, but I don't know how to go about solving it. If anyone has some insight it would be great help. Thanks For every odd integer $n$, $3 \leq n \leq 199$, there exists an integer $m \geq 0$, and a prime…
Trevor Arjeski
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questions about probabilistic primality test

As usual I used online Miller-Rabin test,but there's one thing that i don't understand: when i tested 2500 digit or so numbers it only took 1 or few minutes,but there was few numbers that took an hour or much more, or worse 2 hour 37 minutes (for…
Gary B
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Why was 1 considered as prime years ago?

I've seen on Maths Is Fun that years ago, 1 was considered as prime, but now, it is not. How did this happen? I know that a prime number has only two factors, 1 and itself, and we have 1, which is also itself. Is this why? Tell me what you…
VladInTheTaylor
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show that if $2^n -1$ is prime than n is also prime

How to prove the above statement? Do you have to use Fermat's little theorem where $a^p = a (\mod p)$ I cannot see how to use the above here I tried to factorise $2^n -1 = (2-1)(2^{n-1} + 2^{n-2} + 2^{n-3} +......+1)$ does this help?
zebra1729
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