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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.

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Conjecture on Infinitely Many Consecutive Pairs of Early Primes

An early prime is one which is less than the arithmetic mean of the prime before and the prime after. Conjecture: There are infinitely many consecutive pairs of early primes MY attempt Well, the fact ...
Saucitom's user avatar
-5 votes
0 answers
61 views

The answer to the following question solves twin prime conjecture, please answer carefully! [closed]

I want to know if the union of the equations $A$ to $E$ covers $\mathbb{N}$ fully for sufficiently large $X$ and $Y$ running over non-negative integers $$A = 10XY + X + Y \ \ \ \ \ \ \ \ \ \ \ \...
zakariya ovizadeh's user avatar
-1 votes
0 answers
19 views

Is a sequence of 4 or more consecutive semiprimes possible?

By observing sequences of consecutive integers having the same number of prime factors, I noticed that there never seem to be more than 3 consecutive semiprimes. Does anyone have an idea of what might ...
François Huppé's user avatar
3 votes
1 answer
141 views

Prime number as divisor

I was doing a question and I observed a thing that I'm not able to prove, it follows: For any prime number $n>2$ , there must be only one solution $k=n-1%$ (given that $0<k<n-1$ ) to the ...
Someone's user avatar
  • 41
2 votes
1 answer
80 views

Approximating the Prime Counting Function as $\pi(x) \approx \frac{x^2}{\ln\left(\Gamma(x+1)\right)}$

Approximating the Prime Counting Function as $\boxed{\pi(x) \approx \frac{x^2}{\ln\left(\Gamma(x+1)\right)}}$ Intro________________ In a unrelated topic I was viewing how the mechanical statistics ...
Joako's user avatar
  • 1,384
6 votes
0 answers
75 views

Does iterating this function with primes as fixed points always result in a prime?

Define $f(n) $ as the largest divisor of $n$ smaller than $n$. (undefined at $n=1$) Define a sequence as $$a_1 = k$$ $$a_n = a_{n-1} + f(a_{n-1}) - 1$$ where $k \geq 2$. Note that $a_n = a_{n+1}$ if ...
codebender's user avatar
0 votes
1 answer
30 views

Conjecture on odd numbers producing (semi) primes

While playing around with numbers, I noticed the following pattern: the sum of two consecutive odd numbers added with the product of said numbers produce a prime or a semi-prime. That is, for any two ...
H.T.'s user avatar
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0 answers
36 views

A simple bound on the proportion of primes less than $2^k$.

Fix $k > 1$ and let $P = \{x \mid 0 \le x < 2^k - 1 \text{ and } x \text{ is prime }\}$. Then I want to show that $$ \frac{|P|}{2^k} \ge \frac{1}{2k}. $$ I know how to prove this by using ...
barrel's user avatar
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1 vote
1 answer
80 views

Is the $n$-th prime with remainder $1$ when divided by $4$ always greater than the $n$-th prime with remainder $3$ when divided by $4$? [closed]

Is this rule of prime numbers correct? The $n$-th prime number that leaves a remainder of $1$ when divided by $4$ always greater than the $n$-th prime number that leaves a remainder of $3$ when ...
Nobuyuki Fujita's user avatar
3 votes
1 answer
113 views

Why are there so many primes in $F_{k-1}+F_{k+1}$ when $k$ is odd?

Let $F_{k}$ be the Fibonacci sequence and let $a_k=F_{k-1}+F_{k+1}$. It is observed that there are many primes in the sequence $\{a_k\}$ when $k$ is odd: $F_2+F_4=4$ (even); $F_4+F_6=11$ (prime); $...
Zuriel's user avatar
  • 5,451
2 votes
0 answers
84 views

Inequality between nth prime and a certain integer sequence

I've noticed the following while playing around with primes: Define a sequence as: $$a_1 = 2$$ $$a_n = a_{n-1} + \left \lfloor{\prod_{k=1}^{n-1} \frac{a_k}{a_k-1}}\right \rfloor$$ Define the $n$th ...
codebender's user avatar
0 votes
3 answers
107 views

Prime numbers equal to $K^n - 1$, for $K > 2$ [closed]

A Mersenne prime is a prime number that is one less than a power of two. Thus, it is a prime number of the form $M_n := 2^n−1$, where $n$ is a positive integer. My question is: Are there any other ...
Marino Segnan's user avatar
-1 votes
0 answers
51 views

Are there infinitely many primes p for which either $p−1$ or $p+1$ is squarefree?" [duplicate]

It is not known if there is an infinite number of primorial primes. Are there infinitely many primes $p$ for which either $p−1$ or $p+1$ is squarefree?" I imagine this is an open problem also,...
Adam Rubinson's user avatar
-1 votes
1 answer
141 views

Is $\frac{a^{p}-1}{a-1}$ ($p$ prime) square-free? [closed]

$\frac{a^{p}-1}{a-1}$ should be square-free with all natural $a$. I've been looking for a while but no one online seems to have an definitive answer.
kiet's user avatar
  • 19
-1 votes
0 answers
54 views

Euler's theorem on primes with form $6n+1$ [duplicate]

I'm a little desperate. I'm quite a beginner on the subject of number theory, but be very curious. I have been looking for days for Euler's proof that all $6n+1$ primes can be expressed as $a^2 + 3b^2$...
corto-maltes's user avatar
1 vote
0 answers
94 views

Expected value of iterations until prime number

I came along a reddit post which described an interesting process described by the following set of rules: Start with an empty 'string' of digits Generate a random digit (0-9) and append it to the ...
Levi Rohring's user avatar
0 votes
0 answers
50 views

Prime distribution in triangular number minus x

I'm experimenting with finding sequences of numbers in which prime numbers occur unusually often or rarely. I noticed that the sequence of triangular numbers practically does not contain prime numbers....
j123123 SZT's user avatar
2 votes
1 answer
127 views

Weird NT Question Related with Primes

Find the number of pairs of natural numbers $(k, p)$ with the following properties: 1)$p$ is a prime number 2)$k \leq 2p$ 3)$k^{p-1}|(p-1)^{k} +1$ 4)$k \neq 1$ Now i tried random stuff like: Case 1($p=...
CLASH ROYAL's user avatar
1 vote
0 answers
131 views

Why is Willans' formula "useless"? [duplicate]

Now, there are a couple of questions exactly about this topic already, but none of the answers I've read was satisfying to me. They are all about the fact that Willans' formula is numerically ...
Elvis's user avatar
  • 610
1 vote
1 answer
63 views

Algorithms for Generating Highly Composite Numbers [closed]

Im currently studying highly composite numbers and am interested in algorithms for generating them. However, I have not been able to find suitable literature or resources on this subject so far. Could ...
user1349380's user avatar
-1 votes
0 answers
77 views

$y = [x+1][x-2] + [x + [x + [x-1]]$, if y is a prime number, then find $[x+2y].$ [closed]

Given $y = [x+1][x-2] + [x + [x + [x-1]]$, if y is a prime number, then find $[x+2y].$ My Attempt: I simplified as follows: $y = ([x]+1)([x]-2) + [x] + [x-1] + [x]$. $y = [x]^2 +2[x] -3$. $y = ([x]-1)(...
Chetan's user avatar
  • 73
2 votes
0 answers
124 views

An idea for a continuous nth prime function.

I had a novel idea for a way to generate a continuous version of the $n$th prime function and was wondering if it would work / if anyone could create it or graph it. The idea is: Start with the Taylor ...
NaiDoeShacks's user avatar
0 votes
1 answer
40 views

Calculation of Hilbert symbol over $\Bbb Q_p$ where $p$ is an odd prime

Let $a,b\in \Bbb Q$ be rationals and $p$ an odd prime. I want to calculate the Hilbert symbol $(a,b)_p$. According to https://en.wikipedia.org/wiki/Hilbert_symbol#Hilbert_symbols_over_the_rationals, ...
user302934's user avatar
  • 1,618
3 votes
1 answer
195 views

Smallest "diamond-number" above some power of ten?

Let us call a positive integer $N$ a "diamond-number" if it has the form $p^2q$ with distinct primes $p,q$ with the same number of decimal digits. An example is $N=10^{19}+93815391$. Its ...
Peter's user avatar
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0 votes
0 answers
38 views

Normal Order of Distinct Prime Factor $\omega(n)$

Define $\omega(n)$ as number of distinct prime factors $n$ has, that is if $n=p_1^{a_1}... p_k^{a_k}$, then $\omega(n)=k$. It is commonly understood that normal order of $\omega(n)$ is $\log\log(n)$, ...
spicychicken's user avatar
0 votes
1 answer
34 views

lower bound of $\sum_{n=1}^x \frac{\mu(n)}{n}$

Denote by $\mu$ the Mobius function. Poussin showed that $$ \sum_{n=1}^x \frac{\mu(n)}{n} = O(1/\log x), $$ and there are further improvements since. I wonder what is known about lower bound of ...
mathflow's user avatar
  • 175
5 votes
0 answers
66 views

Can a unique factorisation domain have a largest prime?

Suppose $R$ is a UFD and $(R,\leq)$ is an ordered ring. Is it possible that $R$ has a largest prime element? Below is my attempt so far to answer this myself, though I'm still unsure what the ...
h4tter's user avatar
  • 499
1 vote
0 answers
45 views

How to estimate $n$ so that the product of the first $n$ primes is greater than a given number?

Given $p \in \mathbb{N}$ I want to approximate how many primes $n$ I need so that $$\prod_{i=1}^np_i>4\sqrt{p}$$ with $p_i$ being the $i$-th prime, e.g. $p_1=2,~p_2=3,p_3=5, \dots$ In Schoof it ...
Yvonne's user avatar
  • 11
0 votes
1 answer
107 views

Bezout's identity for Rings

We know that if $F$ be a field and $F[x]$ is a polynomial ring, then $\gcd(p(x),q(x)) = 1\Longrightarrow \exists r(x),s(x)$ such that $r(x)p(x)+s(x)q(x) = 1$. Can we say this same if $f(x), g(x) \in ...
Afntu's user avatar
  • 2,219
4 votes
1 answer
115 views

What is a prime sieve method, and how did they help Zhang, Maynard and Tao?

At children's school we learned about the Sieve of Eratosthenes for sieving our primes from an interval of natural numbers. I was surprised to hear that "sieve methods" were used to make ...
Penelope's user avatar
  • 3,325
0 votes
2 answers
83 views

$ a^2 + p b^2 = c \mod p^2 $ is always solvable?

Let $p$ be an odd prime number. Let $c$ be a given integer between $0$ and $p-1$. It seems that for every $p$ and every $c$ we can find integers $a,b$ such that : $$ a^2 + p b^2 = c \mod p^2 $$ Is ...
mick's user avatar
  • 16.4k
0 votes
0 answers
52 views

Confusion over Riemann's prime counting function

I'm trying to get my head around Riemann's prime counting function, given here (equation 3) as $$f\left(x\right)=\pi\left(x\right)+\frac{1}{2}\pi\left(x^{1/2}\right)+\frac{1}{3}\pi\left(x^{1/3}\right)+...
Peter4075's user avatar
  • 849
0 votes
0 answers
47 views

The prime race $a_i \mod 11$ vs $b_i \mod 11$ conjecture

Let $f(n,a)$ be the number of primes of type residue $a \mod 11$ between $1$ and $n$. Is it true that for all $n>1$ we have $$f(n,1)+f(n,2)+f(n,3)+f(n,5)+f(n,7)+f(n,6)+f(n,8) > f(n,4)+f(n,9)+f(n,...
mick's user avatar
  • 16.4k
-1 votes
1 answer
108 views

Collatz conjecture and prime numbers [closed]

With the intention of understanding how prime numbers contribute to the numerical results we get when we perform any possible numerical calculation (also on real numbers), since they are those natural ...
Matteo's user avatar
  • 45
5 votes
1 answer
152 views

About the solutions of $ \dfrac{x^p - y^p}{x - y} = a^2+pb^2 $

Using theorem $IV$ from this article, is possible to prove that when $p$ is a prime $p ≡ 3\bmod4$, $x ≢ y\bmod{p}$ and $\gcd(x,y) = 1$, then the equation $ \dfrac{x^p - y^p}{x - y} = a^2+pb^2 $ ever ...
user967210's user avatar
2 votes
0 answers
110 views

Is this conjecture about twin primes known to be false?

I'm not sure if this has been investigated before. This is a kind of strong twin prime conjecture Define a first twin prime as the lower of a twin prime pair, while a second twin prime is the upper of ...
Zuhair's user avatar
  • 4,631
0 votes
1 answer
38 views

Counting odd integers in a consecutive sequence divisible by a given prime

For any integer $a, n > 1$, let $O_p(a,a+n)$ be the count of $i$ such that $a < i < a+n$, $i$ is odd, and $p | i$ where $p$ is any odd prime. Does it follow that for any such $p, a, n >1$, ...
Larry Freeman's user avatar
1 vote
0 answers
99 views

Every even number is the sum of at most three primes

I'm failing to find online references to the following problem, which to me seems a slight weakening of the Goldbach conjecture. Conjecture: every even integer $n$ is the sum of at most three primes. ...
CryptoZiddy's user avatar
1 vote
2 answers
72 views

Expected number of factors of $LCM(1,…,n)$ (particularly, potentially, when $n=8t$)

I’m trying to prove something regarding $8t$-powersmooth numbers (a $k$-powersmooth number $n$ is one for which all prime powers $p^m$ such that $p^m|n$ are such that $p^m\le k$). Essentially, I have ...
Lieutenant Zipp's user avatar
1 vote
1 answer
87 views

The sum $f(x)= \sum_{p \le x} e^{\frac{1}{p\log p}}$ is very close to $\pi(x)$. Why is that?

The prime counting function $\pi(x)$ counts the number of primes less than a given $x$. There are other counting functions like Chebyshev's functions which count sums of logarithms of primes up to a ...
zeta space's user avatar
1 vote
0 answers
56 views

What's The Minimum Number Of Prime Factors Needed To Replace "3x+1" With Any Linear ("mx+b") Function And Still Work Like The Collatz Conjecture?

Apologies; I know there are a few assumptions used to pose this question, namely: 1): That yes, any mx+b function can work like the infamous "3x+1," problem... ...Provided, that you give it ...
neuroDiverse's user avatar
1 vote
1 answer
48 views

Generating function of partitions of $n$ in $k$ prime parts.

I have been looking for the function that generates the partitions of $n$ into $k$ parts of prime numbers (let's call it $Pi_k(n)$). For example: $Pi_3(9)=2$, since $9=5+2+2$ and $9=3+3+3$. I know ...
Lorenzo Alvarado's user avatar
0 votes
1 answer
70 views

Are there $a,b$ such that the infinite series of reciprocals of primes of the form $a+bk, gcd(a,b)=1$ is convergent?

My motivation to this question is the divergence of the infinite series of all primes $2$ and primes of the form $2k+1$, so I want to know if there exist $a,b$ such that the primes of the form $a+bk, ...
Mahmoud albahar's user avatar
1 vote
2 answers
162 views

Optimizing Miller-Rabin by selecting bases calculated from the number to test

(First to say, i am not a mathematician, so possibly this is all well known since 1000 years and boring for you. Tell me if so...) I tried to optimize Miller-Rabin prime test by selecting better bases,...
PrimeTester's user avatar
1 vote
0 answers
35 views

A functional equation for a prime divisor finding algorithm's complexity [closed]

I wanted to estimate the computational complexity of a basic algorithm to find all prime divisors of a given number $N$. We'll look for potential divisors for up to $\sqrt{N}$ which would be about $O(\...
Loading - 146 Complete's user avatar
2 votes
1 answer
131 views

Proving there are infinitely many primes using factors of Fermat numbers [duplicate]

Is this proof acceptable? Theorem (Lucas) Every prime factor of Fermat number $F _ n = 2 ^ {2 ^ n} + 1$; $(n > 1)$ is of the form $k2 ^{n + 2} + 1$. Theorem The set of prime numbers is infinite. ...
Pedja's user avatar
  • 12.9k
0 votes
1 answer
58 views

For each integer $k,$ does there exist a $k-$tuple of primes, $(p_n)_{n=1}^{k},$ s.t. for each $n,\ p_{n+1}=2p_n- 1$ or $p_{n+1} =2p_n+1?$

For each $k\in\mathbb{N},$ does there exist a $k-$tuple of primes, $(p_n)_{n=1}^{k},\ $ such that for each $n,$ the following is satisfied: $p_{n+1} = 2p_n- 1\ $ or $p_{n+1} = 2p_n + 1?$ If yes then ...
Adam Rubinson's user avatar
1 vote
1 answer
127 views

Primes of the form $3^a+2^b$ or $3^a-2^b$

I was searching for information about prime numbers, and somehow, I ended up verifying whether certain numbers are prime. I seem to have stumbled upon some far-fetched relationships with the following ...
majster's user avatar
  • 19
1 vote
0 answers
83 views

Determine asymptotically how many positive integers ≤ x are odd, squarefree, and have an even number of prime divisor [closed]

I have known the proportion of the number of odd squarefree intergers to numbers less than x, and I also know the proportion of the number of squarefree integers that have even number of prime ...
鄭賀榕's user avatar
1 vote
1 answer
64 views

Find the next "consecutive-prime composite number" from a given one.

Good day all. I am not a mathematician by a long shot. Please bear with me... I am playing with "descending-consecutive-prime composite numbers" (I don't think that's the term). These are ...
Jaco Van Niekerk's user avatar

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