Mathematics Stack Exchange
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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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If $p\geq 5$ is a prime number, show that $p^2+2$ is composite.

Problem: If $p\geq 5$ is a prime number, show that $p^2+2$ is composite. Remarks: Now if one observes that $p$ takes the forms $6k+1$ and $6k+5$, the problem is resolved quite easily. However, if one were to choose other forms say $4k+1$ and $4k+3$…
Student
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Proof for finite number of truncatable primes

How do we prove that there exists a finite number of truncatable primes? It's intuitive that as it gets bigger it has more factors, so less chance of it being primes when truncated, but what is the mathematical proof?
pbsh
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How to check if a large number is prime?

Is $31!+1$ a prime number? Prime numbers can be written as $6n+1$ or $6n-1$ form. $31!+1$ can also be written as $6n+1$,but not every number of the form $6n+1$ is prime. How do I proceed efficiently?
mac07
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What percentage of prime number factorials plus 1 are themselves prime?

One of the steps in Euclid's proof of the infinity of primes is sometimes misinterpreted to be a way of generating new prime numbers. Specifically constructing the number P!+1 where P is a prime is sometimes though to guarantee a new prime by naive…
matt_black
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Is $1000000000000066600000000000001$ (Belphegor's prime) actually a prime?

There is a Wikipedia article about that evil Belphegor's prime, but the references there seem relatively weak. Is this number actually a prime?
h22
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Find the numbes of a prime arithmetic sequence

I need to find five prime numbers that form an arithmetic sequence with a common difference of 6. I tried to use that 3 will be divisor of the common difference and congruence, but I need to show it in a way that an High school student can…
TAQ
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Is $\lim_{n\to \infty} \frac{np_n}{\sum_{i=1}^n p_i} = 2$ true?

Noob here. I was playing around with primes in JavaScript and I found that if I divide the nth prime times n to the sum of primes up to n, I get closer to 2 for each n going to infinity: $$\lim_{n\to \infty} \frac{np_n}{\sum_{i=1}^n p_i} =…
Marijn
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What is the next prime with this form?

The following are primes, $$P_1 = 2^2 + 3^3$$ $$P_2 = 2^2 + 3^3 +5^5 + 7^7$$ After these two, the only prime of such form I've found is, $$P_3 = 2^2 + 3^3 + 5^5 + 7^7 + 11^{11} + 13^{13} +\dots+ 83^{83} + 89^{89}$$ Is there a prime number with this…
Freddy vs Bieber vs Jason
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Can $2^{M^N}+M^{N^2}$, where $M$ and $N$ are odd primes, never be a prime?

Q: Is number of the form $$\displaystyle 2^{M^N}+M^{N^2}$$ always composite for $M,N$ odd primes? I observed that: If $M=N$ then this number is absolutely a composite, because it satisfies the identity $a+b \mid a^k+b^k$ if k is odd numbers. If $M…
Arnold Karn
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Property of the sequence of primes

Let $p_n$ denote the $n$-th prime number. Does anyone know a proof of the following property? $$\forall n, n', \ p_n p_{n'} \geq p_{n+n'}$$ I'm surprised I can't find anything on this while I believe it should be something people could find…
roger
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Do we really know the reliability of Mathematica's PrimeQ[n] for (n>10^16)?

Possible Duplicate: Do we really know the reliability of PrimeQ[n] (for $n>10^{16}$)? The algorithm Mathematica uses for its PrimeQ[n] function is described at http://mathworld.wolfram.com/Rabin-MillerStrongPseudoprimeTest.html That web page…
Ted Ersek
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More primes of the form $6k-1$ than $6k+1$

Let $a_n:= $ number of primes of the form $6k-1$ and $\le n$ and $b_n:= $ number of primes of the form $6k+1$ and $\le n$. I was playing with my computer and noticed that $a_n\ge b_n$ for all $n\le 10^8$. Can this be extrapolated for larger…
Grobber
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Is $2013^{2014}+2014^{2015}+2015^{2013}+1$ a prime? (usage of a computer not allowed)

Prove or disprove: $$2013^{2014}+2014^{2015}+2015^{2013}+1$$ is a prime number, without using a computer. I tried to transform the expression $n^{n+1}+(n+1)^{n+2}+(n+2)^{n}+1$, but couldn't reach useful conclusions.
VividD
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Bounds on prime number spacing

Suppose n is an integer. What kind of bounds do we know for how close the closest prime p > n will be? I'd especially appreciate an answer that pushes me in the right direction of proving a good bound for this without showing the proof…
Cam
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Do prime numbers satisfy this?

Is this true that $n\log\left(\frac{p_n}{p_{n+1}}\right)$ is bounded, where $p_n$ is the $n$-th prime number?
Papagon
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