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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21 views

Conjecture on Prime Numbers and Exponential Equipment

How to generate a rigorous proof for - The specific conjecture which states that we can always frame every Prime number in an exponentially framed equation with perfect cubes & squares. I'll ...
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Continuation of the Riemann Prime Counting function

Let $f(x)=\sum_{n=1}^{\lfloor \ln(x)\rfloor} \frac {\pi(x^{\frac 1n})}n$ ($\pi(x)$ is the Riemann prime counting function, here are some more informations). It is thanks to Riemann well known that $\...
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Are Wieferich primes Wieferich numbers?

In the article 'On a conjecture of Crandall concerning the $qx+1$ problem' by Franco and Pomerance, they define Wieferich primes to be prime numbers $p$ for which $p^2|2^{p-1}-1$ and Wieferich numbers ...
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1answer
46 views

Is there a set of primes such that the union of their multiples occurs with frequency $1/n$, where $n$ is an integer?

Are there two or more primes, $p_1, p_2, \ldots$ such that the union of their multiples occurs with frequency $1/n$, where $n$ is an integer? If so, can that n be prime? I put it into a computer and ...
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Limit involving prime and composite numbers

Can someone help me to figure out what $$\lim_{n\to\infty} \frac {c_n}n-\frac{c_n}{p_n}-\frac {c_n}{n^2}$$ is equal to? I am pretty sure its $1$ and i tried many different things but i couldnt figure ...
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1answer
40 views

Generalization of digit-wise divisibility lemmas [duplicate]

Background In undergraduate Abstract Algebra homework, for an integer $n$ with decimal representation $a_m a_{m-1} ... a_1 a_0$, I proved that $3$ divides $n \iff 3$ divides $\sum_{i = 0}^{m} a_i$, ...
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Maximum $k$ for odd consecutive primes $p,q$ and $q-p=2k$ for $1$ to $k$ [on hold]

For consecutive odd primes $p, q$ with $q-p=2\cdot k$ for consecutive numbers $1$ to $k$, is some maximum $k$ known?
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How to prove 1111…11 (91 digits) is a prime or composite number? [duplicate]

How to prove $1111......11$ ($91$ digits) is a prime or composite number? My Approach: $1111......11$ can be expressed as $10^{0}+10^{1}+10^{2}+...…..+10^{90}$ Using summation of a geometric ...
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1answer
33 views

Limitations on the difference of coprime composites less than $n^2$ for ascertaining primality?

Let $q$ be a prime where $n < q < n^2$. Let $p_1, p_2, \ldots, p_k$ be all the primes $\leq n$. Is it true that for any $q$, you can find two numbers $a,b$ where $a+b=q$ or $a-b=q$, such that $...
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82 views

Prove $r$ the smallest quadratic non-residue modulo $p \geq 3$ is prime

I've been struggling to find the solution for this question for a while now and thought I might as well ask for some help. The question is: Let $r$ be the smallest positive quadratic non-residue ...
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41 views

$π(x+y) - π(x) ≤ c·y/\ln(y)$ for some constant $c$?

Thinking about the prime number theorem, I wondered whether it is known that there is some constant $c$ such that $π(x+y) ≤ π(x) + c·y/\ln(y)$ for every integers $x,y > 1$. I read that experts ...
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Prime number and Relationship of Sequences of period 4,5,and 6

Let $p$ be a prime number.($p \neq 2,3,5$) Let $t^+,t^-,a$ be sequences. $t^+_{k+5}=t^+_k,t^+_1=0,t^+_2=-1,t^+_3=-1,t^+_4=0,t^+_5=2$ $t^-_{k+5}=t^-_k,t^-_1=-1,t^-_2=0,t^-_3=0,t^-_4=-1,t^-_5=2$ $a_k=...
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Construction of an exact function for counting primes in intervals.

I have constructed an exact function for counting primes in intervals and am curious to know if it 1) has any importance? 2) Has been derived already? I have no formal education in number theory, and ...
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1answer
36 views

Solving a linear equation in one component of $\Bbb{Z}^3$.

Consider the space $X = \Bbb{Z}^3$, a $\Bbb{Z}$-module. Let $M = \{ \sum_{i=1}^n c_i(p_i, q_i, r_i) : \sum_{i=1}^n c_i q_i = 0,$ where $p_i, q_i, r_i$ are either prime numbers or $0 \}$. Then is $M ...
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Prime Zeta function at 1

I wanted to find $$\lim_{s\to 1} (P(s)-\ln(\zeta(s)))$$ and here is my attempt: So we know that $$M=\gamma +\sum_{n=2}^\infty \mu(n) \frac {\ln(\zeta(n))}{n}$$ and that $$P(s)=\sum_{n=1}^\infty \mu(...
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Is there any simplification to:$\mod(P_{k-1}\# \cdot J , \; P_k )\quad$?

Is there any way to simplify or to find pattern in: $$\mod(P_{k-1}\# \cdot J , \; P_k ) \quad with\quad J=0:P_k-1 $$ Sorry for not being clear at notaion, i will try to do it by example: $P_k=...
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117 views

Every sufficiently large positive integer is the average of $n$ distinct primes for certain $n \geq 2$?

I want to generalize a stronger Goldbach's conjecture a little bit because that might help solve it. I was thinking: For all $n \geq 2$, every sufficiently large positive integer $x \geq b_n$ is ...
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Found a new way to do CRT on prime vector

I found a new way to do CRT on prime vector. Given prime list P, and some residuals R=mod(n,P) , for example: P=[2 3 5 7 11] R=[1 0 3 1 9] This matlab function ...
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91 views

Where can I download a full list of all primes below $10^{15}$?

I would like to do some computing on a large list of primes. Unfortunately my computer is not strong enough to quickly generate such a list, so I'm looking to download a file that already contains ...
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Is there a Lay-Mathematician explanation for the proof technique of Zhang's Theorem?

I've heard there are elements of Sieve Theory and whatnot but no further 'outline' of how he proved the theorem.
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An interesting algorithm about prime numbers that I thought today

I thought up the following algorithm today: Choose $a_1\in\mathbb{Z}^+\setminus\{1\}$. Then let $a_{n+1}=a_n+p_n$, where $p_n$ is the largest prime factor of $a_n$. The algorithm is easy, but I ...
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Approximating $r_{0}(n)$ with an integral

I'm still trying to find a tight upper bound for the quantity $r_{0}(n):=\inf\{r\geq 0,(n-r,n+r)\in\mathbb{P}^{2}\}$. My idea is that one should have $\sum_{r=1}^{r_{0}(n)}\Lambda(n-r)\Lambda(n+r)\...
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Is the additive rational group $\mathbb{Q},+$ generated by $\frac{1}{p}$ where p is a prime?

So it is known that the additive group of rationals numbers $\mathbb{Q},+$ is generated by $\frac{1}{n}$ with $n \in \mathbb{N_0}$ so that: $$\mathbb{Q},+ =grp\{\frac{1}{n} | n \in \mathbb{N}\}$$ Now ...
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83 views

Checking for prime constellation

How does one systematically check if a given configuration of prime numbers $p_1, p_2, ... p_n$ is the densest possible configuration of primes in the range $[p_1, p_n]$? (The densest configurations ...
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Are there an infinite number of primes which are any multiple of $n$ apart? [closed]

Are there an infinite number of primes which are any multiple of $n$ apart? That is take $n\in \mathbb{N}$, then is there an infinite number of primes which are separated by $\textbf{any}$ of the ...
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Solutions of $x^n=a$ modulo primes.

Let $p$ be an odd prime and $a$ be an integer that is not a perfect square. We can impose a congruence condition on $p$ that guarantees the equation $x^2=a \pmod p$ does not have a solution. For ...
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Is the following always prime?

For a given $k$ define $$s_k = 1 + \prod_{i=1}^k p_i$$ $$t_k = \text{NextPrime}(s_k)$$ $$v_k = t_k - s_k +1$$ Where $p_i$ is the $i$th prime number. Conjecture: $v_k$ is prime Example: $$k=3$$ $$...
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28 views

Ordering sequences of prime decompositions

C.f. Engel's Problem Solving Strategies; p. 132, #33: Out of $n+1$ integers $\leq 2n$, find $p,q$ s.t. $p|q$. Rather than soliciting a solution I'm wondering if there's a way I can make my attempt ...
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1answer
84 views

Error term of the prime number theorem on GRH?

I know that under GRH we have $$ \sum_{\substack{1 \leq n \leq X \\ n \equiv a (q) }} \Lambda(n) = \frac{X}{\phi(q)} + O(X^{1/2} (\log X^2)). $$ From this I would like to deuce a abound for $E$ ...
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Are there forced factors of numbers of this kind?

Let $$f(n):=n^{n^2}+(n+1)^{(n+1)^2}$$ for a positive integer $n$. For clarification $n^{n^2}$ means $n^{(n^2)}$ , analogue for the other summand. Can we find a concrete factor of $\ f(n)\ $ (like ...
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Complex numbers $z$ such that $n-z$ and $n+z$ are Gaussian primes

Is it known whether for all large enough positive integer $n$, there exists a complex number $z$ such that $\vert z\vert<n$ and both $n-z$ and $n+z$ are Gaussian primes? If yes, are there results ...
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1answer
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Is it true that for all large enough integer $n$, there is a integer $m$ such that $n+im$ is a Gaussian prime?

Let $n$ denote a positive integer greater than $6$. Is it known whether there is always an integer $m$ such that $n+im$ is a Gaussian prime?
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1answer
47 views

Is this sum convergent or divergent? [duplicate]

Is the sum $$\sum_{p\text{ prime}} \frac{1}{p\ln(p)}$$ convergent or divergent ? It is well known that the sum $$\sum_{p\text{ prime}} \frac{1}{p}$$ is divergent but very slow divergent. The above ...
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4 distinct integers with prime sum for each triple

Here is a nice high school olympiad math problem: Can you choose 4 distinct positive integers so that the sum of each 3 of them is prime? How about 5? It looks that just by looking at reminders mod ...
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Use a pairing function for prime factorization

Let $n$ be the product of the two unknown primes $a$ and $b$ ($a < b$): $ab = n$ My idea was to use a pairing function so: $a = f(p)$ and $b = g(p)$ Then the equation is: $f(p) \cdot g(p) = n$ ...
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1answer
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First few smallest Carmichael numbers congruent to $11 \pmod {12}$

There are known to be infinite Carmichael numbers congruent to $a\pmod b$ for coprime integers $a$ and $b$. There are plenty of examples of small Carmichael numbers congruent to $1, 5, 7 \pmod {12}$, ...
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show that $n\Upsilon_{n-1} \equiv -1 (mod\ n)\ \ \ \ \ \iff \ \ n\ is\ prime $

The Bernoulli numbers $B_n$. where all numbers $B_n$ are zero with odd index $n>1$. first values are given by $B_{0} = 1$ , $B_{1} = -1/2$, $B_{2} = 1/6$, $B_{3} = -1/30$. Agoh conjecture: let $...
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1answer
54 views

CRT on prime vector

I have a question that intrigue me: Given primes and some reminder vector P=primes(13)=[2 3 5 7 11 13] R=[1 2 1 3 9 11] (R=mod(1571,P)) What option do i have ...
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1answer
56 views

How dense are primes congruent to 1 and 3 (mod 4)? [duplicate]

There are infinitely many primes of the form $4n+1$ and $4n+3$. In a given interval $[0,N]$ for a large enough $N$ do we expect to see the same number of primes congruent to $1$ and $3$ (mod 4)?
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How many divisors of $\phi(m)$ do not divide $m-1$?

Lehmer's totient problem asks if there exists a composite number $m$ such that $\phi(m)$ divides $m-1$. Lower bounds on $m$ has been established but we do not know if a solution exists. Clearly, if we ...
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1answer
66 views

Confusing thing at the proof of proth theorem

$Proth$ $theorem$ simply depends on a result which proved by pocklington ; The result says : Let $N-1=q^nR$ where $q$ is prime, $n\ge1$ , and $q$ doesn't divide $R$. Assume that there exists an ...
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4answers
219 views

Can anyone come up with an interesting consequence of the Twin Prime Conjecture being true?

The question is in the title. Was wondering if there are statements equivalent to or a consequence of the statement that there are infinitely many twin primes. If not, then why is this conjecture a "...
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1answer
60 views

How many times are these three quantities simultaneously prime?

For recreation, I would like to count the number of times the path length, "area above," and area below the prime counting function, are simultaneously prime. The following function is used to count ...
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78 views

Unique pattern in addition of digits

Problem: For positive integers $n,k$, let $$S(n,k)=\sum_{i=1}^{n}i^k$$ and for positive integers $m,b$, with $b>1$, let $D(m,b)$ be the sum of the base-$b$ digits of $m$. Q$1$- Show that ...
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1answer
73 views

Primes of the form $a^2+b^2+c^2$, $0<a<b<c$

Are there results showing which prime numbers that can be expressed as the sum of three different integers greater than zero?
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109 views

If $n = 18k+5$ is composite, there are at least 9 divisors of $\phi(n)$ which do not divide $n-1$.

If $n$ is a composite of the form $18k+5$, there at least 9 divisors of $\phi(n)$ which do not divide $n-1$. Is this true in general or if not, what is the smallest counter example? Update: No ...
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1answer
39 views

(m,m+2) is twin prime, iff 4((m−1)!+1)≡−m(mod m(m+2))

I'm a programmer, a newbie on math. I'm trying to code to list twin prime. I've found this: $(m, m+2)$ is twin prime, iff $4((m-1)! + 1) \equiv -m \pmod {m(m+2)}$ The pair (m, m + 2) is twin prime,...
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on the proof of AKS algorithm

while reading Primes is in P , I didn't get why can we move from : (X + a)^n = Xn + a (mod Xr − 1, p) (3) (hypothesis) and (X + a)^p = Xp + a (mod Xr − 1, p) (4) (True since p is prime) to (X + a)^n/...
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3answers
522 views

How to make a pair of six-sided dice whose sum is always a prime number?

In other words, how can I find two sets of six distinct integers $a_1, \dots, a_6 \in \Bbb Z$ and $b_1, \dots, b_6 \in \Bbb Z$ such that $a_i+b_j$ is prime for any $i, j \in \{1, \dots, 6\}$?
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33 views

How to prove that $r_{0}(n)/n<1/2$?

Under Goldbach's conjecture, define for a large enough integer $n>n_0$ the quantity $r_{0}(n)$ as $\inf\{r\geqslant 0,(n-r,n+r)\in\mathbb{P}^{2}\}$. Is it possible to prove that $\forall n>n_{...