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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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Are there an infinite number of primes satisfying these three conditions?

(1)   $p$ is prime, and so is $(p+1)/2$. (2)   $p$ can be written as $a^2 + b^4$, $a,b \in \mathbb{N}$. (3)   $p \equiv 2 \bmod 31$. These three conditions were contrived to make a $2017$ puzzle. Now I ask the title question just out of…
Joseph O'Rourke
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Why do we call primes, and not the number one, *the atoms of numbers*?

The fundamental theorem of arithmetic asserts that we can produce every composite number from a unique set of prime multiplicands, so long as none of those primes equals one. Consequently, some mathematicians call the primes the atoms of numbers.…
Hal
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How to prove $ \sum_{k=1}^{p-1}\left\lfloor \frac{k^3}{p}\right\rfloor=\frac{(p-2)(p-1)(p+1)}4 $?

Equation $(36)$ at Mathworld's Prim Sums page reads: $$ \sum_{k=1}^{p-1}\left\lfloor \frac{k^3}{p}\right\rfloor=\frac{(p-2)(p-1)(p+1)}4 $$ I'm curious how this can be proven, but I have no idea...
draks ...
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Primes between $n$ and $2n$

I know that there exists a prime between $n$ and $2n$ for all $2\leq n \in \mathbb{N}$ . Which number is the fourth number that has just one prime in its gap? First three numbers are $2$ , $3$ and $5$ . I checked with computer until $15000$ and…
aeyalcinoglu
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Is $\small \sum\limits_{p\leq x}\pi\left(\frac{x^2}{p}\right)-\sum\limits_{p\gt x}\pi\left(\frac{x^2}{p}\right)=\pi^2(x)$ of any use?

I was playing with some Meissel/Lehmer formulas and I found this one. In fact there is a much simpler way to find it when looking closer, so I guess i is well known. $$\sum\limits_{p\leq x}\pi\left(\frac{x^2}{p}\right)-\sum\limits_{p\gt…
Collag3n
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Is it possible to find the position of a prime number online?

$2$ is the first prime number. $3$ is the second. If I give a prime number such as $1151024046313875220631$, is there any software/website which can give the position of the prime number? I know there are resources to find $N$th prime. But I am…
endrendum
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First 100s place without a prime

What is the first $100$'s place without a prime? $700$ -> primes: $701, 709, 719$, etc. $103900$ -> primes: $103951, 103963, 103967, 103969, 103979$, etc. But at some point, there are gaps of hundreds and thousands between prime numbers, so what is…
Luke Schlangen
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Is this published upper bound on the n'th prime wrong?

This paper gives an upper bound on the n'th prime for $n\ge7022$ as:$$p_n\le n(\log{n} + \log{\log{n}}-0.9385)$$ citing the reference Robin, Guy, Estimate of the Chebyshev theta function on the $k$th prime number and large values of the number of…
BlindAnagram
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Is 2014411 Prime?

I want to figure out whether the number 2014411 is prime or not. What kind of mathematical searching tools/strategy I have? Do I have to start from 1 and proceed to 2014410 while I found one as a divisor or any other efficient way to find it?
Beverlie
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Is 75986⋯69 ever prime or is there a 6 padded prime for every prime $p > 5$?

I came across this question: How can we turn any number into a prime number by simply adding more digits? While trying different approaches to find an algorithm that increases the chances of finding a prime this way I discovered the six padded…
pietfermat
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Largest set of consecutive prime numbers.

This is related to Euclid's proof of the infinitude of primes. For any finite set $S = \{p_1, \ldots, p_r \}$ of primes, consider the number $n = p_1 p_2 \ldots p_r + 1$. This $n$ has a prime divisor $p$. But $p$ is not one of the $p_i$ (where $1…
Buddha
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Is every natural number a prefix of a prime number?

Possible Duplicate: Proof that there are infinitely many prime numbers starting with a given digit string Let n be the representation of a natural number in a non-unary base. Is it a prefix of the representation of a prime number over the same…
Boris Trayvas
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Number we know all prime numbers less than

We already know some very big prime numbers. ($2^{257,885,161} − 1$ as of time of writing is the largest known) It is my understanding, that we know it is a prime number but we don't know all prime numbers smaller than it. An equivalent of this is…
George Padolsey
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What is the ratio of number of prime to number of natural numbers

$\Bbb{P}$ is the count of prime numbers in $\Bbb{Z}$ And so, $\Bbb{Z}-\Bbb{P}=NP$ is the count of non-prime numbers in $\Bbb{Z}$ what is the answer of this equation: $\Bbb{P} / NP$ I thought that question and I made that proof, if I'm mistake please…
errzzz
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Is there a "canonical" representation of integers using numbers other than primes?

Consider the (cumbersome) statement: "Every integer greater than 1 can be written as a unique product of integers belonging to a certain subset, $S$ of integers. When $S$ is the set of primes, this is the Fundamental Theorem of Arithmetic. My…
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