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
11
votes
4 answers

Why does Euclid write "Prime numbers are more than any assigned multitude of prime numbers."

In Euclid's Elements Book XI proposition 20 (http://aleph0.clarku.edu/~djoyce/java/elements/bookIX/propIX20.html), Euclid proves that: Prime numbers are more than any assigned multitude of prime numbers. I know that this is supposed to say something…
Kasper
  • 13,528
11
votes
1 answer

Finding the 10001th Prime

I'm helping my son with Project Euler and we're working on problem 7, "What is the 10001st prime number?" We'll use a Sieve of Eratosthenes and we'll increase the size of the initial array until we're left with 10001 primes. We'll start with a…
uncle brad
  • 374
11
votes
3 answers

Can Prime numbers be negative?

I was wondering, can a prime number be negative? We had a question over at security.se which stated that prime generation with openssl: openssl dhparam -text 1024 results in a 1024-bit number to which leading zeros are added (resulting in 1032…
Lucas Kauffman
  • 211
11
votes
2 answers

can a Car Registration Number, a combination of prime, be prime?

While waiting in my car, I noticed registration number of a car parked in front of my car was 6737. So it was a concatenation of two prime numbers 67 and 37. Now I know following ways to check whether any number is prime or not Let $p$ be the number…
Anurag
  • 283
11
votes
6 answers

To find whether $a$ is a prime number

I have been using this rule to determine whether a number is a prime number, but not how to prove it. Why it has to be $\sqrt{a}$? If $a$ is not divisible by all the prime numbers less than or equal to $\sqrt{a}$, then $a$ is a prime number.
learning
  • 1,749
11
votes
1 answer

Is there a natural number for which all the sums and differences of its factor pairs are prime?

The 8 factor pairs of e.g. 462 are $((1, 462), (2, 231), (3, 154), (6, 77), (7, 66), (11, 42), (14, 33), (21, 22))$. Of the 16 non-negative integers which are the sums and differences of these pairs (such as $462+1=463$, $462-1=461$, $21+22=43$, and…
Andrew Moylan
  • 211
10
votes
1 answer

Is the following always prime?

For a given $k$ define $$s_k = 1 + \prod_{i=1}^k p_i$$ $$t_k = \text{NextPrime}(s_k)$$ $$v_k = t_k - s_k +1$$ Where $p_i$ is the $i$th prime number. Conjecture: $v_k$ is prime Example: $$k=3$$ $$s_3 = 1+(2)(3)(5)=31$$ $$t_3 = 37 $$ $$v_3 = 37-31 +1…
sheppa28
  • 929
10
votes
2 answers

Showing $2^n -1$ is not prime for $n > 2$ where $n$ is even

I'm asked to prove that $2^n -1$ is not a prime number if $n$ is even and $n > 2$. Frankly, I don't know where to begin. So, how should I start tackling this question ?
SpiderRico
  • 483
10
votes
4 answers

Why is $p_n \sim n\ln(n)$?

I know that the prime number theorem states that the number of primes less than $x$ is approximately $\frac{x}{\ln(x)}$. However, why does this mean that $p_n \sim n\ln(n)$? (where $p_n$ is the $n$-th prime). If we replace $x$ with $p_n$ in the…
b_pcakes
  • 1,501
10
votes
1 answer

Mind-boggling property of a prime

As you have already probably known, an Emirp is a prime whose reversal give a different prime i.e: 37 is an Emirp because 37 is prime and its reversal 73 is also a prime, 79 is also an Emirp. Now I have this 100 digit Emirp with a mind-boggling…
Andrew G
  • 103
9
votes
6 answers

Why it is important to find largest prime numbers?

It always takes a lot of effort and money to find the next largest prime number. Why is it so important to do this work and what is the application those numbers?
Mohsen
  • 227
9
votes
3 answers

Multiplying prime numbers

If I multiply $13$ and $17$ to get $221$ I can only get $221$ by multiplying $13$ and $17$ (excluding $1$ and $221$) does the same rule apply to multiplying $3$ numbers? (excluding the use of $1$)
user521254
9
votes
5 answers

How are large prime numbers found?

So I've found multiple ways of determining if a number is prime but I was wondering how they pick a number in the first place. For example the GIMPS project I assume is just trying different numbers to square 2 by but what about other methods? Or is…
Samantha Clark
  • 359
  • 3
  • 6
9
votes
2 answers

What prime numbers have the sum of their digits as a prime number?

(Meta comment: Congrats to Andre Nicolas! I am happy for Andre Nicolas that he is second ranked now. Also he has 3001 answers with no questions. That is good. I am also glad to see Arturo Magidin has been online yesterday.) Now my question is the…
treehugger
  • 91
9
votes
1 answer

Is there a prime which is the form of $10^n + 1$ except $2, 11, 101$?

Is there a prime which is the form of $10^n + 1$ except $2, 11, 101$? I confirm there isn't such prime for $n < 64$.
TOM
  • 1,479
1 2
3
…
60 61