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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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An easy way to count sum of squares of first x prime numbers?

I want to count sum of squares of first $x$ prime numbers. $$1^2+2^2+3^2+5^2+\ldots +p_x^2$$ Is there a formula to do it? (like the formula to count the sum of squares of natural numbers)
Josef Klimuk
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Are these all prime quadruplets?

Wikipedia states on this page that $\{p, p+2, p+6, p+8\}$ is "the only prime constellation of length 4". But I believe this is not true, since $\{p, p+4, p+6, p+10\}$ and $\{p, p+6, p+12, p+18\}$ also satisfy the definition of a prime constellation…
Klangen
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Are all Odd,Non-Prime numbers compulsorily multiple of 3.If yes,how so??

I just thought that if all odd numbers which are non prime, compulsorily multiples of 3,I tried to get some mathematics but in vain. Kindly help me.
Pratyush Kumar
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Is this an efficient type of sieve?

Let $f(n)$ be the nth number of the form $6k\pm1$, which can be defined by $f(n) = 3n + \frac{3}{2} - \frac{(-1)^n}{2}$. Now $f(n)$ obviously outputs all primes $p \gt3$, but also composite numbers whose factors were previously given by $f(n)$. (In…
SmallestUncomputableNumber
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Prime Number Distribution

Lists of prime numbers are often given 'up to' some number. (x primes up to 100,000, y primes up 1,000,000 etc.) Yet their distribution is connected to processes that are frequency-based and thus logarithmic in the same way as musical notes. (Thus…
PeterJ
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How many Primes divid two coprime numbers the same way?

For natural coprime numbers A, B HOW MANY prime numbers P, Q, R, ... are there that: A+B ≡ 0 (mod P) And A+B ≡ 0 (mod Q) And A+B ≡ 0 (mod R) Is there any upper limit or lower limit for those prime numbers?
Nyari
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How fast can we find primes (number of computations needed+time for the computation too)

So, I know we can get a bound on how long it will take to find a large prime. For example, using the fact that between $N$ and $2N$ there must be a prime. And the fact that all numbers between are prime iff they do not have a divisor less than…
picakhu
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Sumbission of answer to CMI

I am very interested in maths and believe to have solved one of the millenial problems. I can understand half of the problems and the other half mean nothing to me but I think I have solved the problem about distribution of primes among natural…
William Pennanti
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Number 2 hasn't got this property, while all prime numbers do.

I am going to start with an example of two geometric figures. Rectangle must haves: quadrilateral four right angles opposite sides are equal and parallel diagonals bisect each other If we say that every square is a rectangle, a square should…
LAAE
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Is it true for all natural number $x$ that $x \le\prod\limits_{p_i\le x} (1+\frac {\ln x} {\ln p_i})$?

I just wonder is this inequality true for all natural number $x$ that $$x \le\prod_{p_i\le x} \left(1+\frac {\ln x} {\ln p_i} \right)?$$ From prime factorization, we have $$x=p_{j_{1}}^{c_{j_{1}}}p_{j_{2}}^{c_{j_{2}}}\cdots p_{j_{r}}^{c_{j_{r}}}$$…
user1992
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Numbers that are NOT prime powers.

Is the set of numbers that are NOT prime powers infinite? I know there is an infinite amount of primes...and therefore an infinite amount of prime powers. I'm just curious if the set if those that are nit prime powers is also infinite. Thanks!
Jon Saboe
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What does the sum of reciprocal of primes less than n equal?

$ \frac 12+ \frac 13+ \frac 15+ \cdots + \frac 1P$ where $P$ is a prime number less than $N$ I have been getting some number approximately $\log\log N$, but I have no nice way of proving it.
Martin Dinev
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asymptotic bound on number of primes from 1 to n

Let $\pi(m, n)$ denote the set of prime numbers in the interval $[m,n]$. I already proved that: $$\prod_{p\in \pi(m+1, 2m)} p \leq \binom{2m}{m} $$ $$\prod_{p\in \pi(1, n)} p \leq {4^n} $$ now, I need to show that - $${|\pi(1,n)| \leq…
Omer.G
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Primality testing

I have this way of testing for primes greater than 2. I divide $2^{n-1} - 1$ by n. If the remainder is 0, n is a prime. This works because in GF(p), if one multiplies all the non-zero elements in the field by the same non-zero number (say 2), all…
Jae Noh
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