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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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Examine if $\exists M$ such that $\forall n>M$, $\pi(2n)$ $-$ $\pi(c_n)$ $>$ $0$

The problem is- Examine if $\exists M$ such that $\forall n>M$, $\pi(2n)$ $-$ $\pi(c_n)$ $>$ $0$. Also find a value of such $M$ for which the theorem is true. Though I haven't still given it a serious thought, but I think that for the second part…
William Hilbert
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Existence of primes $p$ such that all the prime divisors of $p+1$ divide $p-1$

This question recently came up to me in a project and is not taken from a textbook. I would like to know if any characterization of such primes is known from literature. They are seemingly rare but do exist. The first five examples are: $3$, $7$,…
user152139
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Existence of a prime in an interval that is not a linear combination of two specified primes?

If $ n = \left( \frac{p+q}{2 } \right) + p q :p,q \in\mathbb{P}-\left\{2\right\} $ Can we show there exists a prime number $\theta : \sqrt{2n} \leq \theta \leq n $ and $\theta$ is not a linear combination of $p$ and $q$ (restricted to positive…
Brad Graham
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Prime number greater than n

Consider the follwing problem: Given $n$ (in binary) output a prime number $p \geq n$ (not necessarily the first prime number after $n$) Are there better techniques than the trivial one that scans $n,n+1,n+2,...$ until a prime is found? And can we…
Vor
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N-th k-almost prime satisfying a certain condition

Let $G(n,k)$ be the n-th k-almost prime. Prove that for every for every $n \in N$ there exists infinitely many $k \in N$ satisfying $2*G(n,k) = G(n,k+1)$. Source: http://mishabucko.wordpress.com
tesgoe
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Find prime pairs satisfying the equation

Find all ordered pairs $(p_{k},p_{k+1})$, where $p_k$ denotes the $k$-th prime, such that for every $m\ \in \mathbb{N}$ there exists $\alpha \in \mathbb{N}$ s.t. $\Omega(\alpha) = m$ so that $p_{k+1} + 1 = \alpha p_{k}$, where $\Omega(x)$ denotes…
tesgoe
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Power of two startiing with any desired sequence of digits

I heard from a video that there is a theorem that proves that exists a power of two wghose leading or last digits can be any sequence of digits for example your telephone number. Can anyone point me to this theorem please?
Alessandro
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For every integer $n$, does $\{2n+p \mid| p \text{ prime }\}$ contain infinitely many primes?

True or false: For every integer $n$, $\{2n + p \mid p \text{ prime} \}$ contains infinitely many primes? A conjecture, a theorem or just false?
Jeffery C. Niemuth
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a and b are integers where gcd(a,b)=p which is a prime. find gcd(a^2,b^2).

Problem 5 (10 points): Let $a,b$ be integers such that $g.c.d.(a,b) = p$ where $p$ is prime. Find $g.c.d.(a^2,b^2)$. (Original screenshot) I've found that $g.c.d. (a^2,b^2) = p^2$ when using examples for $(a,b)$ like $(9,12)$, $(34,85)$, and…
magnuspaajarvi
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Infinite number of primes in the sequence $1+t^2$?

Possible Duplicate: Primes of the form $n^2+1$ - hard? $1, 2, 5, 10, 17, \ldots$ Are there an infinite number of primes in this sequence $1 + t^2$, $t$ being an integer?
qed
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Number of twisted primes less than 20,000

We call number $N$ a twisted prime if we turn all the 6-es in 9s and all the 9s in 6-es and it remains a prime(If it has no 6-es or 9s it is not twisted). How many twisted primes there are? ($N \leq 20000$) Source: School friend. Is there a way to…
mATHADDICT
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Approximating this integral without using Mertens' theorem

Take $p$ as prime, $\text{li}(x)$ as logarithmic integral and $$ R(x)=\sum_{p\leq x}\frac{\ln p}p-\ln x $$ Without using Mertens' theorem find $$ \int_0^x\frac{tR'(t)}{\ln t}dt $$ I tried using integration by parts and got stuck…
Shivanshu
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Is there a way of showing there are arbitrarily big gaps between primes by contradiction?

This may be a stupid question. So apologies in advance, if it is, One proof of this is a straightforward construction. For any $N$, consider $(N+1)!+2$, $(N+1)!+3$,..., $(N+1)!+N$. All of these numbers are composite. My question is, is there a…
Lost1
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Counting prime powers

The number of prime powers (exponents $\geq$ 2) up to x is given by: $x^\left(\frac12\right)+x^\left(\frac13\right)+x^\left(\frac14\right)+ $...$ =O(\sqrt x$ $lnx) $ (http://mathworld.wolfram.com/PrimePower.html) I am not sure of the $O(...)$ here,…
Navneeth G
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Generating function for the nth prime

Is there a generating function for nth prime that is easy to deal with? i.e. is there a simple closed form for the series $p_1x + p_2x^2 + ...$ or of the form $\sum_{n = 1}^\infty x^{p_n}$
Mayank Pandey
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