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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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Primes in random strings of numbers?

Given any number with random digits, can you always arrange sequential primes in such a way where that number can be formed? Example: $31017643$ is broken into $3,101,7,643$ given all of those are prime. Of course we can eliminate every number that…
TobsterStrudel
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Bertrands Postulate generalization

Bertrands postulate states that there's always a prime number in [N,2N] and I was thinking... Considering that N=1*N and that (1,2) are the first prime numbers maybe this is just a particular case and there's a more general law such as: "There is…
Fabio Beka
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There are at most $n$ primes between $1$ and $2n$

This question originates from one of my tasks: Choose $n+1$ whole numbers $a_1 \le a_2 \le ... \le a_{n+1}$ between $1$ and $2n$ inclusive. Prove that among those $n+1$ number there exist 2 indexes $i$ and $j$ such that $a_i |a_j$, or $a_j$ is…
idkwhatismyname
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Bound on number of zeros in smallest prime greater than $10^n$

I'm inspired by this comment by Eugene Wallingford on a blog post by John D. Cook. Take this Python code, using SymPy (note, I used this branch to ensure that nextprime is not too slow). >>> from sympy import nextprime >>> for i in range(20): ... …
asmeurer
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Prime Numbers and Multiples?

Other than prime numbers are all numbers multiple of 2,3,5 and 7 (Other Prime numbers as well). Suppose like if we need 8 it's the combination of 2.2.2, and 15 as 5.3 etc.
user1467270
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Relationship between $(2^n)^{th}$ prime and its bits

How many bits does it take to write the $p_{2^n}$, the $(2^n)^{th}$ prime number in binary where the first one is 2, second 3, third 5 et cetera. Is it about $n$? E.g., if I take the product $p_{2^{25}}p_{2^{50}} = x$ then will $x$ be approximately…
scribe
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A weird contradiction in semi-primes.

I know that the title and tags doesn't really help, it doesn't really specify what this question is about. But forgive me, I couldn't come up with a better title. I would dearly request the readers to edit the question title and tags to make it…
Rounak Sarkar
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What is $0$ in the concept of prime and composite integers?

All numbers could be prime or composite except $1$. But what about $0$, is it prime or composite?
Stack Exchange
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Prime numbers and variables

What variables can be prime numbers in a following equality: We are working in natural numbers $$b^2=a^2+ca\quad \text {with:}\quad a+b>c, \,c+a>b,\, c+b>a$$ I have managed to prove, that $b$ can not be prime and $c$ can, how to prove that $a$…
King Crimson
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Arithmetic operations on numbers represented by prime factors (optimizing calculations)

Shortly, I'm asking for ideas on how to do 2 things: Given the prime factors of two numbers $x,y$, find the prime factors of $x+y$ without finding $x+y$ and factoring it. Given the prime factors of two numbers, determine which number is bigger, or…
potato
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Formula to get the exact number of primes below $n$

Total noob question so please bear with me. There seems to be several formulae which estimate the number of primes below $n$, but none which give the exact number (short of computing the primes directly). Is that because it has been proven that such…
Fidel
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Is there a function which ranks numbers based on how composite they are?

Say there was a function f(x) which would display integers based on how composite they were, and numbers that had more factors than others would show up as spikes on the graph. Take the number 840. This number has 32 distinct factors, and according…
eners49
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Prime Number Sums

The prime numbers $a,b$ and $c$ are such that $a+b^2=4c^2.$ Determine the sum of all possible values of $a+b+c.$ My Attempt $a+b^2=4c^2$. $a=4c^2-b^2$. $a=(2c+b)(2c-b)$. After this, I tried testing cases but I'm not totally sure how to account for…
R. G.
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Power analogies to primes

As we all know, a natural number $n$ is prime if and only if there do not exist natural numbers $x, y$ exclusively between $1$ and $n$ such that $xy = n$. Is there any generally recognized analogy for primes in powers? For example, a natural number…
waiwai933
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extention of the 6n$\pm1$ prime rule for $30n$, $210n$ ...

I am currently creating a program to exhaustively search for prime numbers. I head from sombody that this rule can be extended by multipliting 6 by the next prime number, giving 30, and then continuing this sequence 210. The offsets from this value…
finlay morrison
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