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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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Let $a, b, c, m, n$ be integers, $m, n$ not both $0$.

(a) Prove that if $am + bn = c$, then $hcf(m,n)|c$ (b) Prove that if $am + bn = 1$, then $hcf(m,(n) = 1$ (c) Prove that $m/hcf(m,n)$ and $n/hcf(m,n)$ are coprime. Question on recent review homework related to coprimes. I know that for a and b to be…
user122661
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Composite Numbers with 1 Prime

What is the method for finding a long sequence of consecutive composite numbers that has only 1 prime? Specifically, how to find 2011 consecutive natural numbers, 1 of which is prime.
Jason Chen
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Is the following statement true

Is the following statement true and how to prove it? \begin{align} (a^2)^{3N} \equiv a^2 \mod{p} \end{align}
SAMARA
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Dirichlet prime counting function?

Let $a$ and $b$ be coprime (i.e. $a \perp b$). Let $f(a,b,x)$ denotes the number of the primes such that $p=ak+b$ and not greater than $x$. For example $f(4,1,10)= 1$. Is there an asymptotic formula for $f(a,b,x)$ which is similar to prime number…
esege
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Numbers used for modeling impose constraints on the model

In modeling observation we use different numbers. Mostly either positive integers or rationals. Both impose constraints on the model description. Positive integers have exactly one minimal element (and are granular). Integers have no minimal…
tesgoe
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Does this function describing super-primes converge?

The prime numbers are positive integers that have no multiplicative structure. One method for counting the number of primes contained in a positive integer is sieving. As an example, for the integer ${10}$, we begin with the first prime ${2}$, and…
user4536
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Will primes always have to solved with brute force?

Its right there in the question. I'm just interested in the subject. Has it been shown that you will always need brute computation to know if a number is prime?
jwillis0720
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Let p be prime and a be an integer such that $p | a^n$ . Is it true that $p^n | a^n$?

I know that $p|a^n$ is $p|(a)(a)(a)...(a)$, $n$ times and therefore $p|a$. I'm not sure where to go from here - could anyone point me in the right direction?
magnuspaajarvi
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$q$ is prime and $k=2,3,4,\dots$ . Show that if $x\in\Bbb Z$ such that $x^2\equiv x\pmod {q^k}$ then $x\equiv 0\pmod {q^k}$ or $x\equiv 1\pmod {q^k}$

Considering the or condition in this statement, would I be proving both could be true or two separate cases? Also, is there some sort of reduction required? I know there exists a formula for $x^2\equiv x\pmod {q^k}$ but something about the exponent…
user122661
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Prove that $a^2 \equiv b^2 \pmod q$ if and only if $a\equiv\pm b \pmod q$ for any prime numbers $q$.

Prove that $a^2 \equiv b^2 \pmod q$ if and only if $a\equiv\pm b \pmod q$ for any prime numbers $q$ homework question, please help.
user122661
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Greatest possible number of primes between 1 and $x$

Background: I'm making a program in java to calculate all the primes between 1 and any given number ($\pi(x)$). I want to create an array that will contain all the numbers, but resizing an array takes time. The solution I came up with was to…
The Guy with The Hat
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Interesting inequality using primorial

Not assuming PNT, what is $a$ in $$(p\#_x)^a=(2^{1/2})(3^{1/3})(5^{1/5})...$$ where $p\#$ is primorial till $x$, and r.h.s is over primes. Also answer can be asymptotic !
user3058477
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Determining if a number is prime

I'm working on a program that determines if a number is prime or not. This involves dividing this number by all numbers smaller than it. But because we are dealing with very large numbers. Dividing it with too many numbers isn't a good idea . So I…
Mohammad
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RSA Questions (again)

The RSA values that state require factoring of two primes, does this mean that "the RSA number" can only have two prime factors but may have other factors that are not prime as well? Is the fastest known algorithm to determine if a int1024 value is…
Derek James
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RSA and finding the factors

What is the biggest hurdle with finding the prime factors for RSA encryption? If 3 doesn't factor, wouldn't that leave 33.3% left, then 20 then 14 as you move along the known more simple primes? and of those, you would only be interested in primes,…
Derek James
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