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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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Is there a possibility for two different primes to have any of its powers to be the same?

Say there are two primes $P_1$ and $P_2$ where $P_1 \neq P_2$. Is there a possibility for some $m$, $n$ ($m \neq 0, n \neq 0$) such that $P_1^m = P_2^n$.
Navneeth G
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Prove if prime can be written as $2^n+1$, $n = 2^k$

Prove that if prime can be written as $\ 2^n + 1$ then $n = 2^k$, $\;\;n, k \in \mathbb N$. I am pretty new in this part of math.
mkostelac
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Proof that the first reappearing remainder when dividing one by a prime number is one

I am trying to proof that the first reappearing remainder when dividing one by a prime number is one. What I found is that if the expansion of $1/p$ recurs with period $k$ then $10^k-1$ is divisible by $p$. What I don't see is how this relates to my…
vonjd
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Are Undulating Primes Endless?

Are undulating prime numbers(literally, prime numbers that can be represented as abababab...aba, such as 919 or 131) endless or limited? Is this problem improvable?
염도현
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Stategy for prime factorization

How do I prime factorize big numbers, such as 8435674686325652 without having to make millions of divisions?
user30523
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Show $p^2 + pq$ is not a perfect square for primes $p,q$

I have the following proof: Suppose it was a perfect square, then $\exists k$ such that $p^2 + pq = k^2$. We can rewrite this as $pq = k^2 - p^2 = (k - p)(k + p)$. Since $p$ is a prime number, it must divide either $(k - p)$ or $(k + p)$. However,…
Jeff
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What way is the fastest to count up the prime numbers in your head(like: 2, 3, 5, 7, 11 and so on for an infinite amount)?

I am a person who gets bored all the time, so I found that counting something in my head helps occupy my time. I started counting in binary, but after a while this started getting old as I was doing it in the back of my mind, so I got bored again.…
Etscharntmin
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Prime powers satisfying $2^n \pm 1$?

Is there any solution to the equation $$ p^k = 2^n \pm 1 $$ with prime $p > 3$ and natural number $k > 1$? For $p=3$, is there any solution except the trivial case $3^2 = 2^3 + 1$? What I did: Look into wikipedia and google on Mersenne primes (half…
Jeroen
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An arithmetic problems

"Find all the positive integers $n$ satisfying $2^n+n^2 +25 = p^3$ where $p$ is a prime number." I proved that $n$ must be divisible by 6 and predicted $n=6, p=5$. However, I can't wrap it up. Thank you for all.
Noun
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Modulus of a large prime, a smaller prime multiplied by a positive factor n doesn't repeat.

Given a prime p1 and a prime p2 (where p2 < p1), I made the observation that with a number n (where n increases by 1 from the value 1) the equation below results in two properties: x = (p2 * n) mod p1 The first repeat is the value p2 where n = p1…
Sean Parsons
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Is there a Non-prime Number that is Divisible only by Numbers Greater than its Square Root?

Possible Duplicate: Calculating prime numbers The question is in the title. Is there a number that is divisible only by numbers greater than its square root? If not, why? I need this because it can speed up a calculation algorithm significantly…
TimeCoder
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Does the probability of finding a violation of Goldbach's Conjecture go down as larger numbers are considered?

In particular, I realize that with each number we check there is slightly less of chance of finding a violation, but instead I am asking if larger numbers are somehow 'padded' with more solutions? And by going higher up in bigger numbers does the…
Juel Herbranson
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How to investigate whether number is prime

Investigate whether 36611 is prime. Show the first few steps of the procedure. This is the question. Is it enough to show its not divisible by 2, 3, 5 ? Surely this must not be enough. But how do i do it ? can some one help please ?
emil
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In the context of prime numbers, what does $(p-1)/2$ signify

I'm looking at some prime generation code which first tests $p$ for primality, then has the option to check $(p-1)/2$ for primality. The code already tests $p$ directly (testing against some known primes and using Miller Rabin), but then has the…
ubiquibacon
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Looking for polynomials that generate primes for integers within a given interval

Euler famously discovered the polynomial $f(n) = n^2 + n + 41$, which generates prime numbers for $n = 0,1,2,3,…,40$, but not $n = 41$ ($41^2 + 41 + 41$ is obviously divisible by $41$). Hearing of this result made me curious to see if there was a…
Mailbox
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