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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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Absolute Prime numbers in base 10.

A prime number is called an absolute prime if every permutation of its digits in base 10 is also a prime number. For example: 2, 3, 5, 7, 11, 13 (31), 17 (71), 37 (73) 79 (97), 113 (131, 311), 199 (919, 991) and 337 (373, 733) are absolute primes.…
Math Tise
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The proof that there are infinitely many primes for 4n + 3 and 6n + 5

I'm trying to prove that there are infinitely many primes for $6n + 5$. I see the same proof for $4n + 3$, and can't understand why we say that all of the primes are either in the form $4n + 1$ or $4n + 3$. If I apply it to $6n + 5$ they will be…
E. Shcherbo
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probability that a large integer N is prime is less than a small integer M - 2 answers?

I want to ask "is the probability of N being prime smaller than the probability of M being prime, if N and M are randomly chosen, and M < N?" Having tried to do some research, as a non-trained mathematician, I find many people ask "what is the…
Penelope
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Are all $30n + 7$ numbers prime numbers where $n$ is a whole number?

So I was lying on my bed and I began to wonder about prime numbers and as I went through each prime number I found that numbers $37, 67, 97, 127, 157, \ldots$ all happen to be prime numbers (correct me if I am wrong). Is there a way to…
anant virk
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On the properties of prime number

Let $p,q$ be two distinct prime numbers such that $p^3 \mid q+1$. If suppose that $p \neq 2$, then can we say anything about $p,q$?
Little girl
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Probabilistic indication of infinitude of Mersenne primes

It is an open question as to whether there are infinite Mersenne primes. However, by the prime number theorem the probability that a large random number $z$ is prime is approximately 1/ln($z$). We can apply this to the Mersenne numbers ($z$ =…
Thomas Delaney
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Odd $n$ as sum of three primes

It is a known result of Vinogradov that every sufficiently large odd integer $n$ can be expressed as sum of three primes (this has been improved recently for all $n\ge 7$, cf. here). However, I do not know whether there is any result regarding the…
Paolo Leonetti
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What is the correlation between $n$ and $n$-th prime?

It looks like the gap between $n$ and the $n$-th prime is increasing as $n$ gets bigger. Is there any function describing this increase? If I would like to find the $n$-th prime with the sieve of Eratosthenes, what is the smallest array I can use?
KOB
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prime gaps 4*n+1 or 4*n+3

Take the two prime gaps of $4*n+1$ and $4*n+3$ and the first ten million primes. Will we find that the number of gaps for each is about the same? One could see the possibilility of a prime gap race similar to that for primes of the form $4*n+1$…
J. M. Bergot
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Where can I validate $300\ 000$ digit prime number is valid one?

I recently found a different method to compute prime number in $\mathcal O(\log(\log n))$ complexity. At present, that logic working fine for $300$ digits prime number, which I found on websites.I need to validate whether that logic will be working…
ideano1
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41 +primorial is prime

I noticed that $41$ plus primorials is prime when I add $2$ to $41=43$, then $6$ to $41= 47$, then $30+41=71$ all the way up to $41+9699690=9699731$. Hence I can add the first $8$ primorials to get primes. Do you know if there is a different prime…
J. M. Bergot
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Sum of prime remainders is greater than prime p

This something like Waring meets the primes. Take some prime p(n) and divide it by all the primes less than one-half p(n) to find the sum of the remainders that are themselves also primes. Is there some sum of these remainders for p(s1) that is…
J. M. Bergot
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A conjecture involving the sum of the squares of prime factors

$x$ is an integer greater than $1$. Consider the following equation: $$(x+1)^2=x^2+p_1^2+p_2^2+\dots+p_n^2$$ where $$x=p_1^{a_1}p_2^{a_2}\dots{p_n^{a_n}}$$ Find all such numbers $x$ satisfying the above equation. So far, $6$ is the only known…
user406287
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Given primes p and q is there guaranteed to be some $\alpha$ and $\beta$ such that $p^\alpha = q^\beta + 1$

Given a prime p and a prime q, is there some $\alpha$ and $\beta$ such that $p^\alpha = q^\beta + 1$ or $q^\beta = p^\alpha + 1$? I am unable to formalize a proof for this, but it seems to me to be true. My reasoning is that, WLOG say $p > q$ ,take…
Andrew Butler
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How to achieve a perfect square

Given a natural number $n$, compute the smallest number $k$ such that $nk$ is a perfect square. Obviously, I have tried factorizing the number so $n=p_1^{s_1}...p_r^{s_r}$ and $k=p_{i1}...p_{il}$ where $p_{ij}\in\{p_m:m\in\{1,...,r\}:s_m\text{ is…
Álvaro G. Tenorio
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