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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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A sketch of proof of Prime Number Theorem

I'm looking for some sketch of the elementary proof of the Prime Number Theorem, which would suffice to explain someone the general mechanism of proving PNT without going into deep analytic methods etc.
user263286
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$(m, m+2)$ is twin prime, iff $4((m-1)! + 1) \equiv -m \pmod {m(m+2)}\ $ [Clement's theorem]

The Wiki page on Twin Primes says The pair $(m, m+2)$ is twin prime, iff $4((m-1)! + 1) \equiv -m \pmod {m(m+2)}$. This is obviously connected to Wilson's Theorem. Can anybody provide a proof for that, along the lines of thought given…
draks ...
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Formula for composite numbers

I was digging around blogspot when I came upon an old post that claimed the author discovered a formula that generates all odd composite numbers. The post: http://barkerhugh.blogspot.com/2012/05/composite-number-formula.html?m=1 The formula: $2x…
Euler77
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Contradicting $p|n $s.t $p > \sqrt n$

I have proved a basic theorem in prime numbers, If $n \ge 2$ and $n$ is composite, then it is divisible by some prime $p \le \sqrt n$. This is a fairly basic result, and then my textbook shows me how the 'Sieve' works etc. But I am interested to…
user214138
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Pythagorean Triplets & Prime Factors

Read this knowing that I have no mathematics background whatsoever. I was solving a specific programming problem that required knowledge of Pythagorean triplets, which is something that I hadn't really worked with before. Anyway, I'm curious if…
clem
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A question about amicable numbers

http://en.wikipedia.org/wiki/Amicable_numbers I'm doing a research on amicable numbers and I wanted to write $p$, $q$ and $r$ numbers of Thābit ibn Qurra theorem. I tried to write $p$ and $r$ in terms of $q$. I calculated $p$ as $\frac{q-1}{2}$ and…
Mehmet
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If I want to learn to count in another base to calculate primality, which one should I try?

I've heard base 12 is better, but what about base 30 ? Learning multiplication tables in another base would be quite fastidious, so I don't know if that already been tried before... (I don't know what sort of tags I should put on this one)
jokoon
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Prime numbers and divisibles

I was wondering that in the process of checking if a number is a prime number, would it be reasonable to suggest that if it cannot be divided by 2, 3, 5, 7 or 9 then it could be considered a prime number? My maths is not the greatest so I am going…
DaveMcK
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Square in Interval of Primes

Denote by $a_n$ the sum of the first $n$ primes. Prove that there is a perfect square between $a_n$ and $a_{n+1}$, inclusive, for all $n$. The first few sums of primes are $2$, $5$, $10$, $17$, $28$, $41$, $58$, $75$. It seems there is a perfect…
William Yi
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Squares of primes in the sequence 7 + 6k

I was bored one day, and I stumbled across the fact that for the first few primes p (p >= 7 at least), have the fact that p^2 = 7 + 6k for some integer k. I couldn't prove it for all primes p >= 7 quickly (made a few stabs in the dark, but no luck),…
Mike K.
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Can you prove this formula for computing prime numbers is correct?

Sometime ago I discovered the following function for computing primes: $$ Q(x)=\text{frac} \left (\cfrac{\Gamma(x)}{x} \right )\cfrac{x^2}{x-1}= \begin{cases} x & \small \text{if $x$ is prime} \\ 0 & \small \text{otherwise} \end{cases} …
user72430
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Are there infinitely many prime numbers?

I will post my own answer below. That should not deter others from answering. There are many ways to prove this.
Michael Hardy
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Prove that 1 is the only "common" divisor of the integers n and n+2

Let n be any odd integer. Prove that 1 is the only "common" divisor of the integers n and n+2. I think you have to find gcd(n, n+2) and say that since n odd then then n+2 will also be odd. Thus n + 2 is either an odd prime or odd non-prime. If n is…
Pasie15
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How to prove that this Proth number cannot be a prime number? (without using a computer)

Without using a computer prove that this Proth number cannot be a prime number : $$43373\cdot 2^{49822}+1$$
Pedja
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Relative error in prime number fitting

I got curious about what it would look like if I made a scatter plot of the n-th prime number as a function of n (lets call it p(n), so that p(1)=2, for instance ). Not being an expert in the area I somewhat expected it to be all over the place, but…
user135206
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