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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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Prime number question?

If I want to calculate all the primes, could I do this? Let $N = \{2,3,4,5,6,7,8,..\}$ Then the elements in $N - NN$ are prime? Because they are the ones that aren't composite?
user247247
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Where can I find a list of large prime numbers

A repository of say 13 digit prime, 15 digit primes etc.
Onkar Singh Gujral
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Euclid's theorem: Paul Erdős's proof on the infinitude of primes

Seemingly simple question: Quote from Wikipedia: First note that every integer $n$ can be uniquely written as $rs^2$ where $r$ is square-free, or not divisible by any square numbers (let $s^2$ be the largest square number that divides $n$ and…
Trademark
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Find the smallest prime positive integer in each of the following.

Find the smallest positive integer such that $80-n$ and $80+n$ are prime numbers. Find the smallest positive prime number such that $2002-n$ and $2002+n$ are prime numbers. I cannot think of any way other than trying the prime numbers one by…
Jacky
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Prove, that the number of prime numbers between $p_n$ to $p_{2n}$ is $n-1$

I observed this phenomenon and checked this up to some(very small) $n$, but it seems so astoundingly trivial and consistent. I'm up for any proof/counterexample and opinion, thanks. Sorry, everyone that I made several mistakes in quoting the…
user18724
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distribution of gaussian primes

here is a naive question that so far I don't have already found somewhere else. In the following, I consider the norm on gaussian integers with $N(a+ib)=a^2+b^2$. Consider prime gaussian integers whose norm is $>2$. For any of them, say $p$, find…
Kongman
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Prove that $p \ge 5$ is prime, then the remainder of $p$ upon division by $6$ is $1$ or $5$.

An example in my textbook, but I'm not quite sure how to set this one up, because of the $p \ge 5$ part. How do I start it off?
JCMcRae
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If $p > 5$ is a prime number, then the last digit of $p^4-1$ is $0$.

If $p > 5$ is a prime number, then the last digit of $p^4-1$ is $0$ (ex.: $7^4-1=2400$). How do I prove this?
user176791
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Prime larger than a twin prime

Wondered whether the following equation holds true for all twin primes such that where $a$ and $b$ are twin primes and where $b=a+2$, then $3\left[\left(\frac{a+b}{2}\right)^2-1\right]+2 = NP$. Where $NP$ is a prime number? I noticed while playing…
anovice
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The Prime Numbers Set is infinite. Is this proof correct?

Proposition: The Prime Numbers Set is infinite. Proof: Suppose we have a finite set of prime numbers $p_{1}, p_{2}, ..., p_{n}$ such that $p_{n}$ is the largest of them. Define $ c := p_{1}*p_{2}*...*p_{n}$ $c$ is clearly not prime. Let $q = c +…
Guilherme Duarte
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Algorithm to identify complex Mersenne primes?

I am thinking on the complex analogue of the Mersenne primes. I think, some like a "complex Mersenne prime" could be a complex prime in the form $$2^{a+b\frac{pi}{2}i}-1$$ Where $a+bi$ is a complex prime as well. Is it an "usable" extension in the…
Milkman
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Iterated Pi function

Does anyone have any information on iterating the prime counting function. Specifically, $\pi_n(x)$=$\pi(\pi_{n-1}(x))$, and $\pi_1(x)$=$\pi(x)$. I'm looking for anything on this function, what it may be called (when I search for iterate pi…
vukov
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Are there infinitely many primes of the form $p_1\cdot p_2\cdot...p_n+1$?

Possible Duplicate: Is there an infinite number of primes constructed as in Euclid's proof? The question is : Are there infinitely many primes of the form $p_1\cdot p_2\cdot...p_n+1$? ($p_k$ is the $k$-th prime.) For example: $2\cdot3 + 1$. But…
titusfx
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Is this proof about the form $2^n \pm a$ correct?

I want to prove following statement : For prime numbers $p$ greater than $3$, it is true that: $a)$ if $p=2^n-a$ and $a=6k+1$, then $n$ is an odd number. $b)$ if $p=2^n+a$ and $a=6k-1$, then $n$ is an odd number. $c)$ if $p=2^n-a$ and $a=6k-1$,…
Pedja
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Is ${(prime^2-1) \over 24}$ always a member of the generalized pentagonal number set?

I was working through a puzzle on why the square of a prime minus one is always a factor of 24 (http://puzzles.nigelcoldwell.co.uk/fifteen.htm) and noticed that the sequence of numbers for ${(prime^2-1) \over 24}$ always seems to be a member of the…
slipheed
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