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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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Proving that there exists a certain set of equations

This may be a dumb question, but it bothers for quite a while. Lets say, we have a certain equation, like $ab-a$ where $a, b$ are primes. Then we generate a sequence for every $a$ and $b$ which looks like $2,3,4,5,6,7,8,10,11,12,13,14,17,...$ Is…
Are
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math theory- primes problem

I have tried to prove it for over $15$ hours with no success. I got a clue to use the following technique: between $((P_n), \:2(P_n))$ there is an additional prime hiding there - $P_{n+1}$. please only clues and no solutions! Thanks.
Stav Alfi
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Mersenne prime is a pseudoprime

Suppose $M_p=2^p-1$, where $p$ is a prime, is composite. Work: Then $M_p = ab$ for positive integers $1 < a,b < M_p$. I want to show $2^{ab}=k2^{2^p-1}+2$ ---> $2^{2^p-1}-2=k2^{2^p-1}$ ---> $2^{2^p-1} \mid 2^{2^p-1}-2$ ---> $2^{2^p-1} \equiv 2…
user292908
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4 Primes in 9 consecutive Integers

This is not so much a question, as it is an interesting (to me at least) observation. Part of a more complex problem involved finding primes in 9 consecutive Integers greater than 2. Not being the smartest math guy out "there", I used my current…
Ole Drews Jensen
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checking if a number is a prime

I was reading Wikipedia, and it was given that "all primes are of the form 6k ± 1" (other than 2 and 3), where k = 1,2,3,4,... Is this statement correct? If yes, can we use this to test if a given number is a prime number? For instance, we can say…
user263229
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Proof on Prime Numbers

Show that for any prime numbers p, q, r , one has the sum of p^2 and q^2 is not equal to r^2 How do i go about doing this question? Do I use mathematical induction or is there any other way to go about doing it? if i use mathematical induction, how…
user271716
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Show a prime number exists between $n$ and $n!$

All $n$ bigger than $2$ has $p$ (prime number) such that $p$ divides $n$. Then a $p$ exists which divides $(n - 1)!$ With this, i have no idea how to show that $p$ is bigger than $n$ ($n < p < n!$).
user34625
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Function generating all primes?

As far as I know, no one has been able to find an onto function $f:\mathbb{N} \rightarrow P$ where $P$ is the set of all primes. Does there exist an onto function $f: \mathbb{N}\times \mathbb{N} \rightarrow P$ where $P$ is the set of all…
Fawkes4494d3
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Breaking a number into two composite numbers.

Prove that every natural number $n$ greater than or equal to 12 can be written as a sum of two composite numbers. Clearly when the number is even, it can be written as a sum of two even composite numbers, but what about when $n$ is odd?
user247855
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Unique Factorization Property

Every Composite number can be factorised into primes in only one way,except, for the order of primes. Except for the Order of Primes?? Please Clarify!
Amit Jha
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Besides 1 and 11, is $\sum_{i=0}^n 10^i$ composite for every $n\in \mathbb{N}$?

Given a number consisting of digits all equal to 1 in base 10 and not equal to 1 or 11, is it necessarily composite? I know that 11 is the smallest non-trivial counter-example, but I would like to know if it is the only counter-example. If not, what…
vosov
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How many number id divisible by $p$ and is not divisible by any primes number which is less than $p$?

Let $N$ be a big integer. Let $p$ be a prime number. Is there a formula to count how many number less than $N$ such that they are divisible by $p$ and not divisible by any prime less than $p$. For example, $p = 3$, one…
GAVD
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What is the smallest "prime" semiprime?

(All dots here means concatenation.) Let $s= ab$ be a semiprime number, then I call s a "prime" semiprime if all these following conditions are satisfied: The reversal of $s$ is a prime The concatenation of $a$ and $b$ in any order(i.e. $a{.}b$ and…
Non-Been being
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$a^4+b^4+c^4+d^4 \neq 2^{2011}$

Prove (elementary, meaning no high level theorems used) that there can not exist 4 prime numbers a,b,c,d $\geq$ 7 such that \begin{equation}a^4+b^4+c^4+d^4=2^{2011}\end{equation} I tried the following: The last digit l(d) of the fourth power of a…
Alexandru
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Is there a systematic way of referring to a prime?

Is there a systematic scheme for identifying primes? For small numbers, it is easy to simply reproduce the whole prime, but for larger numbers, it seems like it could get cumbersome. For instance, one could instead refer to the "$n$th" prime, but…
Superbest
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