Mathematics Stack Exchange
Mathematics Stack Exchange

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
3
votes
0 answers

Some question of the theorem of infinitely many primes

Here is the Theorem in my textbook: For each nonconstant polynomial $f(x)\in\mathbb{Z}[x]$, the set of prime divisors of the integers $\{f(k):k\in\mathbb{N}_{0}\}$ is infinite, where $\mathbb{N}_{0}=\mathbb{N}\cup\{0\}$ and…
user1992
  • 1,366
3
votes
1 answer

Property of prime integers mod 3

Looking at primes mod 3 starting at 5 we get a sequence 2 1 2 1 2 1 2 2 ... Counting the number of 1's minus the number of 2's we get a second sequence -1 0 -1 0 -1 0 -1 -2 .. . Then from the first sequence we derive a third sequence 1 2 1 2 1 2 2…
Stephen Wynn
  • 111
3
votes
1 answer

Root of prime numbers

Possible Duplicate: use contradiction to prove that the square root of $p$ is irrational I was sitting at school bored, and I suddenly thought about prime numbers and an interesting question popped up in my head: $$\bf\text{Is the root of every…
JohnPhteven
  • 2,007
3
votes
0 answers

prime number function.

Let $f$ be a following function : ${ f: \mathbb{N} \rightarrow \mathbb{P} }$ ${ f(n) = p_n }$, where $p_n$ is n-th prime number. I want to prove or find a proof that ${ f \in \Theta\left(n\log\left(n\right)\right) }$ (using Big-Theta notation) which…
nonamenick
  • 31
3
votes
2 answers

For what primenumbers, $p$, is $\sqrt{5p +49}$ an integer?

For what primenumbers, $p$, is $\sqrt{5p +49}$ an integer? I managed to figure out: $5p +49 = n^2$ $5p = (n+7)(n-7)$ But can't think of anything more than that, anyone able to solve this without bruteforcing? For primenumbers under $10^7$, the…
user485593
3
votes
0 answers

Size of the $k$-th prime number

Is it possible to give an estimation (maybe an approximation) of the size of the $k$-th prime number?
Adam54
  • 301
3
votes
3 answers

Is $21300113901613$ prime?

Is $21300113901613$ prime? And is there any easy way to check? I'm just checking properties of primes from the sequence $a(1) = 3$ and $$a(n) = \frac{a(n - 1)^2 + 1}{2}.$$
user460386
3
votes
0 answers

How high do we know for certain all the prime numbers?

Where do the first gaps start appearing in what numbers we know are prime or not? For example this website http://compoasso.free.fr/primelistweb/page/prime/liste_online_en.php has a list of all the primes up to 1 trillion. How much higher could you…
Samantha Clark
  • 359
  • 3
  • 6
3
votes
2 answers

Number squared minus 1 never prime?

It seem like if you take a number and square it and subtract 1 it is never prime. Of course if you take an odd number and square it and subtract one you get an even number.I have been making my own table of primes and so far the squares -1 are not…
Charles Vossbrinck
  • 31
3
votes
1 answer

Simplifying logarithms for inverse of the prime counting function

I am writing a prime number sieve in software. I want to be able to return the first $n$ primes, where $n$ is an argument to the sieve function. In order to do this efficiently, I would like to be able to allocate an array of integers that I know…
user171305
3
votes
1 answer

Question about sum of primes

Let $P(x)$ be the xth prime, $P(0)=2$ Given a prime $p$, does it always exists a $n$ such $\sum_{x=0}^{n}P(x) = 0$ $(mod$ $p)$ ? Example : $p=7$ $2+3+5+7+11=28=4*7$ Best regards
BenLaz
  • 807
3
votes
2 answers

Prime powers that divide a factorial

If we have some prime $p$ and a natural number $k$, is there a formula for the largest natural number $n_k$ such that $p^{n_k} | k!$. This came up while doing an unrelated homework problem, but it is not itself homework. I haven't had any good ideas…
AnonymousCoward
  • 5,548
3
votes
1 answer

How to find primes between given range

I saw movie "the man who knew infinity" In which ramanujan had formulated a formula to calculate number of prime numbers between a range of numbers My teacher told me that there exist no program which can generate primes My friend told me that…
super saiyan
  • 344
3
votes
3 answers

Can I use PNT in this way?

I want to show that for large $n$ the $n$-th prime grows like $n\ln (n)$. Is this correct? By PNT $$\mathop {\lim }\limits_{x \to \infty } \frac{{\pi (x)\ln (x)}}{x} = 1.$$ Let $x = {p_n}$, so that $\pi (x) = n$, where ${p_n}$ is the $n$-th prime,…
glebovg
  • 10,154
3
votes
1 answer

Generating pairs of primes from the 2 previous primes.

Is every odd prime of the form $\,p+q\pm1\,$ for primes $\,p,q?\,$ This question arose as below. First of, i have no idea if this is the right place to dump this "potential finding", but i have found no other forum or journal where i could submit…
bbuubbi
  • 41
1 2 3
…
60 61