# 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.

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### How many number of ordered pair $(m, n)$ can be formed if $m+n=190$ and $m$ and $n$ are positive integers and coprime?

The question involves the concept of number theory Kindly provide the hints to solve the question not the entire solution I don't know how to approach these kind of problems
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### Problems implementing the Meissel-Lehmer method

I'm working on an implementation of the Meissel-Lehmer method for calculating $\pi(x)$. I've figured out how to recursively compute $\phi$, but I'm having trouble with a more basic portion of the code....
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### Is there a way to analytically extent $x^2+x^3+x^5+x^7+…+x^{prime(n)}+…$

Is there a way to analytically extent $x^2+x^3+x^5+x^7+...x^{prime(n)}+...$ I was wondering because I know that for $|x|$ smaller than $1$ it converges. I have been wondering for 2 years now I haven't ...
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### Congruential equidistribution in infinite sets or sequences of positive integers

Let $S$ be an infinite set of positive integers, $N_S(z)$ be the number of elements of $S$ less than or equal to $z$, and let $$D_S(z, n, p)= \sum_{k\in S,k\leq z}\chi(k\equiv p\bmod{n}).$$ Here $\chi$...
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### Towards a new proof of infinitude of primes ( with possible unified application to other primes of special forms whose Infinitude is unknown):

I'm trying to prove the infinitude of primes as follows: Consider the following partial sum : $$S(p)=\sum_{n=2}^p\sin^2\left(\frac{π\Gamma(n)}{2n}\right)$$ The summand is zero for non-primes greater ...
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### Properties of prime sum graphs

The prime sum graph $P_n$ on the vertex set $V(P_n) = \{1,\dots, n\}$ has an edge $e = xy$ when $x+y$ is prime. It is easy to show that any such $P_n$ is bipartite (put odd numbers in one part and ...
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### Can you prove that a seeming growing sequence goes to infinity?

I was given this problem from my brother he told me to prove that a sequence goes to infinity. It starts at 21. You write 21 in hereditary base 2 notation like this $2^{2^2}+2^2+1$. He told me to ...
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### how to establish one to one correspondence between prime set and natural number set? [closed]

i know Fundamental theorem of arithmetic and prime-counting function $\frac{x}{ln\,x}$, but it can't help establish one to one correspondence.
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### what is the probability that you get a prime number when you pick two random numbers between 0 and 1 ,a and b, and divide b by a and round up?

what is the probability that you get a prime number when you pick two random numbers between 0 and 1, a and b, and divide b by a and round up? let's say your random numbers where 0.09121=a, and 0.6163=...
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### what is the probability that a prime number divides another prime plus 1?

what is the probability that a prime number divides another prime plus 1? what I do know is that for 2 it's 100% I can show this fact using a function $f(x,y):=$ the number of primes between $1$ & ...
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### Why is -1 not a prime number? [duplicate]

I understand the reason 1 is not considered to be a prime number, but what is the reasoning for -1 not being considered a prime number? It's only factors are 1 and itself, -1, wouldn't that make it a ...
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### Understanding principles in integer sequence

I am an engineering student and when I was doing some work on data visualisation I stumbled across an integer sequence after watching a video about sequences that produce interesting graphs. I ...
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### On the inequality $\left(\frac{R_{n+1}}{R_n}\right)^n<n^{\frac{5}{4}}(\log n)^3$ for Ramanujan primes

The Wikipedia Firoozbakht's conjecture refers (see also the comments of OEIS A182514) an inequality due to Nicholson, I wondered if it is possible to prove the following conjecture. Conjecture. The ...
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### Prime number logic

It is conjectured that for every intever $n\geq1$ there is a prime $p$ with $n^2<p<(n+1)^2$. Show that if this conjecture is true then $\pi(x)\geq\lfloor\sqrt{x}\rfloor$ for all $x\geq2$. I ...
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### Counting consecutive integers that are divisible by primes relatively prime to an arbitrary $n$

Let: $c > 0, n, m, x > 0$ be an integers $p\#$ be the primorial of $p$ $D_n(m,x)$ be the count of integers $i$ where: $m-x \le i < m$ There exists a prime $p$ that $p \nmid n$ but $p | i$ ...
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### Factorial modulo a larger prime

$\textbf{Question:}$If $n$ is an integer and $p$ a prime larger than $n+1$, are there any conditions we can put on $n$ so that $n! \equiv -1 \pmod{p}$
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### Existence of primes less than double a previous prime (two steps back) - extending Bertrand?

Say we have three sequential primes: $p_1, p_2, p_3$ I know by Bertrand's postulate that $p_2<2p_1$ However, what I wonder is for $p_1>7$ if the following is always true: $p_3<2p_1$ A ...
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### Relationship between primes, right triangles and homogeneous polynomials

It is known that if $x^2 + y^2 = z^2$ is a primitive Pythagorean triplet then $z$ is not divisible by any prime of the form $4k-1$. The following is a generalization of this classical result which ...
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### Infinite prime proof using factorial plus one or product of primes plus one?

My instructor provided a proof for the Theorem: The number of primes is infinite.Proof by ContradictionAssume finite number of primes this means there is a largest prime say $p$.Now lets say there is ...
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### Using elementary methods to prove infinitely many primes mod n

I was reading an elementary number theory text looking to enhance my knowledge and I came across the relatively simple task of proving there existed infinitely many primes of the form $4k-1$ (of ...
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### How many numbers between 1 and 1,000 (both inclusive) are divisible by at least one of the prime between 1 to 50? How can I find this? [closed]

I was trying to solve a compettive programming problem in which constraints are so high so I want to deduce a formula for it so that i could do it for other ranges as well.
By Zsigmondy's theorem, there are infinitely many prime divisors of $2^{2^n}-1$. That is, the set $$A=\{p \text{ is a prime}: p\mid 2^{2^n}-1 \text{ for some }n\in\Bbb{N}\}$$ is infinite. Also, as ...
### The equation $y^2=x\pm \ell$
For what odd primes $\ell$ does the equations $y^2=x\pm \ell$ have a finite set of solutions over the integers. Here, assume $y$ is even and $x$ is a prime number. I am not sure if this is really ...