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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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Proof that $2^n-1$ does not always generate primes when primes are plugged in for $n$?

Exactly what the name entails. The function $2^n-1$ I see largely tends to generate primes when $n$ is prime. However, a week ago I heard that this was horribly false. Please show me a disproof.
AAron
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Are 7 and 49 coprime?

Or 6 and 36, 5 and 30, and things like that. They aren't, right? A co prime is a pair of numbers whose greatest common factor is 1. They (7 and 49) share 7 as well as 1. If 7 and 49 aren't co prime, it will clear up a huge programming question.
Ben
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Is it correct to say a number $n$ is prime if $n \bmod a \neq 0$ for $2 \leq a \leq\sqrt n$?

As I was playing around with Fermat's little theorem, I came up with another method to check if numbers are prime, if the remainder of the division of $n \over a$ was not $0$ for any integer a between $2$ and $\sqrt n$, then the number should be…
user530873
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Are the first 1,000 prime numbers enough to build every Goldbach number up to 9 digits long?

I'm writing a basic computer program in which one of my requirements is to find the smallest pair of prime numbers that make up a Goldbach number (up to 9 digits long, non-inclusive). The user inputs any positive even number and I want the program…
nofe
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How to determine if a number $A$ is divisible by all the prime factors of $B$?

How to determine if a number $A$ is divisible by all the prime factors of $B$? For example: $120,75$ $A=120=2^3\times3\times5$ and $B=75=3\times5^2$ Therefore yes, $A$ is divisible by the prime factors of $B$.
Aniruddha Paul
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The name given to the number 1 in the context of Primes and Composites

We give names to the sets of numbers called Primes and Composites. Is there a name for the number 1, in this context, seeing it is neither a Prime or Composite?
Paul
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A miraculous number N

Of course we can talk about 1 digit prime numbers, 2 digit primes , 3 digit primes , and so on..., my question is : is there an N (N greater than zero) such that there are no N-digit prime numbers ? I realize that it will be truly miraculous if…
Arnold C
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Determine if $n$ is prime?

If $n < 10^6$ and no integer between $1$ and $10^4$ divides $n$. Is n prime? Here is my attempt: Assume $n$ is prime. Then using trial division, $n$ must be divisible by an integer between $1$ and the square root of $10^6$ (which is $1000$). Thus,…
Peter
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Prime numbers in a formula

I am not able to give a proof of the following statement: given an integer number $k$, we consider the following expression: $$x=\sqrt{k^3}-\sqrt[3]{k^2}$$ Show that you can get infinite prime numbers from this formula. I tried for $1\le k \le 1e7$…
Riccardo.Alestra
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Maier's theorem

I have some questions with Maier's theorem If $1 < \lambda < 2$, then what? If $x+(\log x)^\lambda = x^{1+1/\pi(x)}$, then what? In particular, if $\lambda = 0.525\log x/(\log\log x)$, then what? (see A result, due to Baker, Harman and Pintz in…
John Nicholson
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Relative primes

What is the number of integers between 1 and 60 that are relatively prime to 60? I know that the answer is 16, but how do I go about finding the relative primes using a quick process?
Hector
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How to compute Dirichlet densities?

I need to calculate the Dirichlet density of the set of primes $p$ of the form $p = n^2 +1$ (in fact show it is zero), but I have no idea how to go about it. My definition of Dirichlet density of a set of primes X is: $dens(X)= \lim_{s\rightarrow…
MattBurrows
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Patterns in Prime numbers, and the null hypothesis

I've read about many attempts to find patterns in prime numbers. First, is there a mathematical way to prove there is not a pattern to prime numbers? Since there are ways to check if a number is prime or not, can these methods be combined into…
Everett
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Odd Primes Problem Proof

Given the odd prime numbers, Prove that if $x$ and $y$ are adjacent odd primes in this list, then $x + y$ has $3$ prime factors. The factors need not be distinct. Here is an example I have provided: $3 + 5 = 8 = 2 \cdot 2 \cdot 2$. Therefore, $8$…
Alexis Dailey
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