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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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15 views

Prime counting function $\phi(x)-c(x)$ vs. $x/\ln(x)$

So $\pi(x)$ is the prime counting function. That is to say, it counts the number of primes below a given integer $x$. This function is very important in number theory. I was wondering how well the ...
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1answer
24 views

How does this method work to find prime numbers?

I'm curious about this pattern that I saw while adding many powers of two together, and then taking the prime factorisation of each result, and I'm curious as to why this occurs. Here is the pattern: ...
2
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1answer
45 views

For every $x\in\mathbb{N}$ write $2x=2\prod_{i=1}^{n}p_i$ with $p_i=6k_i+1$ primes

I want to prove the following: $$\forall x\in\mathbb{N}\ \exists\text{ a prime }p\equiv1 \pmod 6 \text{ s.t. }p|(2x)^2+3;$$$\ \text{i.e. } (2x)^2\equiv-3 \pmod p$ for some $p\equiv1 \pmod 6$ has to ...
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2answers
11 views

Are mersenne primes of exponent being a mersenne primes always prime?

A mersenne number is a number on the form (2^n)-1 . For it to be prime, the number n must be prime. My question is that if n it's another mersenne prime, will 2^n-1 be always prime? It seems so, to ...
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0answers
36 views

Compositeness tests for numbers of the form $k \cdot b^n \pm c$

Can you provide proofs or counterexamples for the following claims? Claim 1 Let $P_m(x)=2^{-m}\cdot((x-\sqrt{x^2-4})^m+(x+\sqrt{x^2-4})^m)$ . Let $M= k \cdot b^{n}-c $ where $k,b,n,c$ are ...
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1answer
34 views

Can the set of odd primes be decomposed into $\Bbb{P} = A + B, $ for some $A,B \subset \Bbb{Z}$?

Can there ever exist infinite sets of integers $A, B$ such that $A + B = \{ a + b: a \in A, b \in B\} = \Bbb{P}$? Where $\Bbb{P}$ is the set of odd primes? You can include $0$ and / or $\pm$ odd ...
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2answers
38 views

“Although it is necessary for $n$ to be prime in order for $R_n$ to be prime” as logic statement

This is a statement about Repunits from this paper. How can I write this as an if/then statement? Knowledge about Repunits isn't required. The question is basically: what does "although it is ...
2
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1answer
29 views

Can you determine small primes from larger primes?

Suppose you are given the primes in the range $[n,n^2]$. Is there a known way to effectively reconstruct the primes less than $n$? Ideally, something that takes less calculation than figuring them out ...
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2answers
44 views

Prove (in)equivalency: (1) $\exists$ non-zero $z\in\mathbb{Z}/n\mathbb{Z}$ such that $z^{t}=0$ for some $t$, (2) $\exists$ prime $p$ such that $p^2|n$

I need to prove the equivalency/inequivalency of two statements: (1) There exists a $z \in \mathbb{Z}/n\mathbb{Z}$, $z\neq{0}$, with $z^{t} = 0$ for some $t \in \mathbb{N}$ (2) There exists a prime ...
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0answers
28 views

Pythagorean and the number 17 [on hold]

I was looking for some properties of the prime numbers, and I found some curiosities of the number 17 in that site. One of them is this 17 was called by the Pythagoreans - opposition, obstruction,...
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1answer
42 views

Prime numbers that are sums of Prime numbers themselves [on hold]

what is the minimum prime number that is the sum of exactly two odd prime numbers? i.e I want to find a counter example to: $$p_i+p_j \in \mathbb P \operatorname{iff} i=1 \lor j=1$$
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1answer
103 views
+50

Compositeness tests for generalized repunit numbers

Can you provide proofs or counterexamples for the following two claims: First claim Let $P_m(x)=2^{-m}\cdot((x-\sqrt{x^2-4})^m+(x+\sqrt{x^2-4})^m)$ . Let $M_p(a)= \frac{a^p-1}{a-1} $ where $a$ ...
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0answers
36 views

Does Goldbach conjecture imply the twin prime conjecture ?

Under the Goldbach conjecture, define a primality radius of a large enough composite positive integer $ n $ to be a positive integer $ r $ such that both $ n-r $ and $n+r $ are primes. Assume $k $...
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1answer
65 views

Finding the n-th prime number [on hold]

We want to uniquely map hash values to prime numbers. One way to achieve this is storing the first $l$ prime numbers into an ordered list $L$ with size $|L| =l$. When the hash value $h$ calculated, ...
6
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1answer
79 views

Prime divisor in positive integers sequences

I would like to know if anyone has an ideea if the following statement is true. For any sequence of consecutive positive integerers $(n_0, n_0+1,..., n_0+k).$ Where $n_0 \ge 1, k\ge 0,$ but $k\ge 1$ ...
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1answer
54 views

Certain Complex Integral

I was trying to generalize the Riemann's prime number formula for $\pi(x)$ to a general algebraic field $K$, and came across the integral: $$f(x)=\frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty}\frac{d}{...
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2answers
27 views

Partial part of fermat's little theorem [on hold]

Am I fool? I cannot understanding the partial part of the proof. $\{1,2,...,p-1\}\equiv \{a,2a,...,(p-1)a\}\ (mod\ p)$ Why above statement is true? How I remove a??
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0answers
17 views

How to efficiently find the number of coprime pairs such that none of the pair elements belong to the same group?

I have 'n' number of positive integers. I want to find the number of coprime pairs such that none of the pair elements belong to the same group. For example, if my number sequence is ...
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0answers
30 views

Prove that any polynomial with integer coefficients must have a composite number in its image. [duplicate]

Let $f \in \mathbb{Z}[x]$ be a polynomial of degree at least $1$. Prove that there is $n \in \mathbb{Z} $ such that the corresponding polynomial function $f(n)$ is not a prime. I think that the ...
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0answers
41 views

Generating prime factors of a certain congruence?

I'm aware that prime factors of $n^2+1$ take the form $4k+1$. It's also well known that factors dividing $\frac{a^p \pm 1}{a \pm 1}$ will be congruent to $2kp+1$. Fibonacci and some other recurrence ...
2
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1answer
32 views

Primes congruent to $k \pmod p$ between $p$ and $p^2$?

Is it true that for any prime $p$, there are primes $< p^2$ which are congruent to every $k<p$? For example, $$ \begin{align} 11 &\equiv 1 \pmod 5 \\ 7 &\equiv 2 \pmod 5 \\ 13 &\...
6
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1answer
140 views

Particular base-10 digit patterns in Collatz

Note: I will use the abbreviation RCF for the Reduced Collatz Function. The arrangement of certain specific digits produce a particular pattern on the next iteration of the Reduced Collatz Function. ...
3
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2answers
54 views

Primes of the form $p_1p_2\dotsm p_k+1$ [on hold]

Let $p_1,p_2,...,p_k$ be the first $k$ primes and $k>1$. Does there exist a prime $p_{k+1}$ such that $p_{k+1} = p_1p_2...p_k + 1$?
4
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3answers
47 views

Proof that Cardinality of Primes is Cardinality of Natural Numbers

I need to prove that the cardinality of the set of prime numbers is the same as the cardinality of the set of natural numbers. This is the proof I came up with: $\mathbb{P}\subseteq \mathbb{N}\...
0
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1answer
23 views

Prove set of primes is equal to set of natural numbers

I was studying for an upcoming test in college and was looking at an old test. I'm struggling to understand how to prove this problem and was hoping someone could help me out. Prove that |P| = |N| ...
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0answers
29 views

How many primes are there on the form $100\cdots 0 1$? [duplicate]

For example 11 and 101 are primes, but apart from them, can we determine how many primes on the form $100\cdots 00 1$ there exist (in decimal number system)?
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1answer
46 views

Prove that Nth prime number is bigger or equal than 3n [closed]

Let $P_{n}$ be the nth prime number. Prove that $\forall n \in N, n \ge 12: P_{n} > 3n$. Tried to use induction here but couldn't get any reasonable relation between $P_{n}$ and $P_{n+1}$.
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5answers
166 views

Is it true that $n!$ divides $p^n(p+1)(p^2+p+1)\cdots(p^{n-1}+\cdots+1)$?

Let $p$ be a prime number (though I suspect that this might be true for composite ones as well). Define $$f(p,n)=\frac{p^n(p+1)(p^2+p+1)\cdots(p^{n-1}+\cdots+1)}{n!}$$ where $n$ is a positive integer. ...
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0answers
12 views

Big-O approximation of probability that the GCD of two large integers is not $\beta$-smooth

For a given $\beta$ and $n$, define $f(\beta)$ as the limit, as $n$ goes to infinity, of the probability that $\gcd(u,v)$ has a prime divisor larger than $\beta$, computed for uniformly random ...
1
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1answer
33 views

RSA signature verification with a different modulus

I got a question on RSA signature verification. To verify, we calculate $y^e$ mod n (where y = $x^d$ mod n being the signature) to get x and compare with the original message. Is it possible to ...
9
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2answers
171 views

Can $pk+1$ divide $(p-k)^2$?

Let $p>3$ be a prime, $0<k<p$. Then is it possible that $pk+1\mid(p-k)^2$? For $k=1$, since $(p-1)^2\equiv(-2)^2\equiv4\pmod{p+1}$, and $p>3$, this is not possible. And if $pk+1\mid(p-k)^...
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1answer
59 views

Question regarding $C(n)$ and $B(n)$ in Hanson's proof that $\prod\limits_{p^a \le n} p < 3^n$

In Denis Hanson's proof, he defines two terms: (1) $B(n)$: which is the Least Common Multiple of $\{1,2,\dots,n\}$ $$B(n) = \prod\limits_{p^a \le n}p$$ (2) $C(n)$: an integer $$C(n) = \dfrac{n!}{\...
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0answers
28 views

Is Dirichlet's theorem on arithmetic progressions true for ring of Gaussian integers?

Is Dirichlet's theorem on arithmetic progressions true for ring of Gaussian integers $\mathbb{Z}[i]$? By Dirichlet's theorem I mean the fact that if some line $an+b$ ($a, b\in Q$ and $n\in\mathbb{Z}$ ...
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5answers
71 views

The relation between irreducible and primes

Am I right in thinking that the conventional definition of a prime integer (can only be written as itself times $1$ and has no other factors) is actually the definition for irreducible? Is it true ...
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0answers
31 views

consecutive prime gaps and explicit bounds

I am aware of the theorem that $p_{n+1} - p_n \leq n^{0.535}$ which is true for all sufficiently large numbers due to Baker, but if i want to make the implicit for all sufficiently large number to ...
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1answer
80 views

Does the following inequality always hold true?

$$ 0\lt \frac{\sum_{i=n}^{n + P_n - 1} P_i}{P_n \cdot P _{P_n}} \leq 1 $$ Or is there a lower bound bigger than zero? Which I believe not to be the case. Some basic examples are as follows: $(1)$ ...
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4answers
82 views

Find the values of $p$ and $q$

If $p^3+p=q^2+q$ where $p$ and $q$ are prime numbers, Find all the solutions (p, q) I tried to solve this exercise using that: $p^2 = -1(\text{mod} \, q)$ and $q = -1(\text{mod} \, p)$; So: $...
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1answer
55 views

Show that if $ p\mid n$ then $\phi(np)=p\phi(n)$

The question: Let $n\in\Bbb N$ and let $p$ be a prime. Show that if $ p\mid n$ then $\phi(np)=p\phi(n)$. What I know is: It is related to Euler's totient function $\phi$ $\phi$($n$)= # of $+$...
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3answers
151 views

Divisors of $\left(p^2+1\right)^2$ congruent to $1 \bmod p$, where $p$ is prime

Let $p>3$ be a prime number. How to prove that $\left(p^2+1\right)^2$ has no divisors congruent to $1 \bmod p$, except the trivial ones $1$, $p^2+1$, and $\left(p^2+1\right)^2$? When $p=3$, you ...
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1answer
135 views

Can the exact number of twin primes $\leq n$ be proved using a “twin-prime zeta function”?

Let $\pi(n)$ denote the amount of primes $\leq n$ and let $\pi_2(n)$ the equivalent for twin primes. Properties of $\pi(n)$ can be proved using a well-known formula involving the zeros of the Riemann ...
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0answers
72 views

Are there infintely many primes generated by the recursion $c_{n+1} = \lceil \frac{3}{2} c_{n} \rceil$?

Inspired by a recent discussion (How to solve a ceiling expression or recurrence equation?) I stumbled on the question: Are there infinitely many primes in power ceiling series? If not there must ...
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2answers
241 views

Numbers of the form $m^2+2^n$

I would like to show that there are infinitely many primes $p$ such that every number is congruent to $m^2+2^n$ modulo $p$ for some positive integers $m,n$. This is what I have tried so far: if 2 is ...
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2answers
45 views

Interesting numbers $n$ such that $x^n-1=(x^p-x+1)f(x)+pg(x)$

I'm dealing with the test of the International Mathematics Competition for University Students, 2011, and I've had a lot of difficulties, so I hope someone could help me to discuss the questions. I'...
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0answers
20 views

Sequence of consecutive integers [duplicate]

I want to prove the following: Let $m_0,...m_r$ be pairwise coprime integers . Show that there exists a sequence of consecutive integers $s, s+1,...,s+r$ such that $m_i\vert s+i, i =0,...,r$ I know ...
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4answers
45 views

Prove that $2^q+q^2$ is divisible by 3 where $q$ is a prime and $q\geq5$. [duplicate]

I'm looking to prove that $2^q+q^2$ is divisible by $3$ where $q$ is a prime such that $q\geq5$. I know that primes greater than five will be congruent to either $1\ (\text{mod}\ 3)$ or $2\ (\text{...
1
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0answers
27 views

About $ \beta_a^b(n) = \sum_{i=2}^n (-a)^{\omega(i)} \space \omega(i)^b $ where $\omega$ is prime omega function.

Let $\omega(n)$ count the number of distinct prime factors of the integer $ n > 2$. This $\omega(n)$ is called the prime omega function. Consider $$ \beta_a^b(n) = \sum_{i=2}^n (-a)^{\omega(i)} \...
1
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2answers
31 views

Invertible elements $\mathbb{Z} / p^2 \mathbb{Z}$

Let $p$ be a prime. We are curious about the invertible elements of the quotient ring $\mathbb{Z} / p^2 \mathbb{Z}$. What we do know is that according to the Euler totient function there are $\phi (p^...
2
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0answers
43 views

Having trouble with Lemma 3 of Hanson's proof that $\prod\limits_{p^a \le n} p^a < 3^n$

I have been going through Hanson's proof (page 33-37) and I was following it up until Lemma 3. There, Hanson does a magic trick which I am not clear on. Here is the step: $$\frac{(n/a_i)^{n/a_i}}{((...
1
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0answers
38 views

Efficient Reduction of $q\mod k2^n+1$?

In the Lucas Lehmer Primality Test, the following identity is used: $q\equiv (q\mod 2^n)+\lfloor\frac{q}{2^n}\rfloor(\mod 2^n-1)$. This allows the modulus operation to converge with only addition ...
3
votes
4answers
36 views

Divisibility in polynomial equations

Let $p$ be a prime number. Let $a, b$, and $c$ be integers that are divisible by p such that the equation $x^3 + ax^2 + bx + c = 0$ has at least two different integer roots. Prove that c is divisible ...