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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.

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does there exist a prime such that...

Let $n>1$ be an non square positive integer (you can have it prime, if you wish), does there exist a prime $p>2$ such that $n$ generates the multiplicative group of $\mathbb F_p$? It sounds true, but I could not find an immediate proof for that...…
Reyx_0
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Proving $\lim_{n \to \infty} \frac {\log{p_n}} {\log n} = 1$

How do I show: $$\lim_{n \to \infty} \frac {\log{p_n}} {\log n} = 1$$ where $p_n$ is the $n$th prime number without using the Prime Number Theorem? Some context: The reason I can not use the PNT (or at least the form one might try to use) is because…
user85798
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My date of birth as DDMMYYYY is a prime number, is it common?

My date of birth as DDMMYYYY is a prime number. I have a bit of a thing for prime numbers so I thought that was pretty cool. I wondered whether that's really special. I already figured that less than $50\%$ of the people have that because even birth…
Ivo
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What's the smallest known interval containing at least one prime number?

Wikipedia says that Dusart proved in 2010 that there's at least one prime between $x$ and $\left(1 + \frac{1}{25\ln^2x}\right)x$ for $x \geq 396738$. For $x_0 = 396738$, this implies a prime between $x_0$ and $x_0+96$. My question is: Is Dusart's…
user263286
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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 repeated/written $n$ times, for…
Kazuya Kazunaga
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why only square root approach to check number is prime

Why do we use only square root approach to find a number is prime or not? why not cube root & 4rth root?
manaswini
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Prime number in a polynomial expression

Will be glad for a little hint: let x and n be positive integer such that $1+x+x^2+\dots+x^{n-1}$ is a prime number then show that n is prime
Myshkin
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Number of lucky primes

The "lucky numbers" can be constructed with this sieve. The red ones are the lucky numbers. As you can see, some are prime. Is the number of lucky primes infinite? Edit: Apparently this is an open problem, so I won't get a proof. However, I do have…
Jimmy360
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Number theory, prime numbers

prove that for all prime numbers $p>2000$ the sum $$1+2^{2000}+3^{2000}+...+(p-1)^{2000}$$ is divisible by $p$ I tried this with primes smaller then $2000$, but I can't find a general rule.
Maria Amarandei
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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 $n \geq 1$, set $M_n$ to be the set of all $m$ such that…
Alejandro Saldivar
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Proving the following false: $f(x)=2x+1$ produces a prime number if and only if x is prime

$f(x)=2x+1$ produces a prime number if and only if x is prime, how can we prove this false? I know this isn't very math-proofy, but can't we just plug in a number we know that is prime, i.e. $x=7$, and observe $f(7)=15$, which is not prime? I'm…
Jason
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Show $x^{\pi(x)} < 3^x$ using the PNT.

Using the Prime Number Theorem show that: $$x^{\pi(x)} < 3^x$$ for sufficiently large $x$. I started off by taking the $\log$ of the inequality such that: $$\log(x^{\pi(x)}) < \log(3^x)$$ so $$\pi(x) \log(x) < x \log(3)$$ by PNT we know that…
sarahusher
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is $324+455^n$ ever prime

Another question that I can only solve in part. Is there an $n$ such that $324+455^n$ is prime? When $n$ is odd, this is false since $$ 324+455^n = (2\cdot 3^2)^2+(5\cdot 91)^n \equiv (-1)^2+(5\cdot 15)^n \equiv 1+(6\cdot 3)^n \equiv…
user144248
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Sum of two primes

In how many ways can $10001$ be written as the sum of two primes? Obviously since the 10001 is odd, one of the primes must be $2$. This leaves the second, must be prime as 9999, but it isn't, hence there are $0$ ways to write $10001$ as the sum of…
jessica
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Proving that this sequence is strictly increasing

Let $C_k$ be the largest positive integer that cannot be written as the sum of $k$ distinct primes for $k\geq 4$. Using that all odd integers greater than 17 can be written as the sum of 3 distinct primes (which I believe is proven in Harald…
SFA
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