# Tagged Questions

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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### A question on the prime number theorem as presented in the following paper

In the section 2. of this paper it is written that, ...The prime number theorem ensures that we can choose $B$ as close to $1$ as we want, provided $x_0$ is sufficiently large. I think that ...
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### Irrationality of the concatenation of decimal expansions of primes

I've heard the proof that this number is irrational is accessible to even a novice to number theory: $\alpha = 0.2 \ 3 \ 5 \ 7 \ 11 \ 13 \ 17 \ldots$ The proof may utilize that a number is ...
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### Periodicity over the prime indices of a multiplicative sequence implies periodicity?

I have a real sequence $(a_p)$ indexed by the prime numbers which takes values -1, 0, or 1, having the property that $a_p=a_q$ whenever $p\equiv q \pmod m$, where $m$ is a fixed integer $>2$. I'm ...
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### Asymptotic Expression for the Twin Prime Counting Function

A variation on a previous question I asked, which has garnered no responses. I'll attempt to be more lucid: Let $\pi_2(x)$ be the twin prime counting function and $\pi(x)$ be the prime counting ...
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### How can one find a million of consecutive prime numbers greater than 1 trillion? [duplicate]

I am looking for bigger prime numbers than 1 trillion. At least a million consecutive ones. Where or how can I find some?
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### A Generalization of Carmichael Numbers

Obviously, from Fermat's Little Theorem, the condition of $p$ being prime is equivalent to there being some number $a$ of multiplicative order $p-1$ mod $p$. Moreover, this is equivalent to saying ...
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### Proof for divisions that in include prime number. [duplicate]

How do I prove, that if $m^2$ can be divided $p$ (where $m$ is a whole number and $p$ is a prime number) then also m can be divided by $p$?
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### Why can't prime numbers satisfy the Pythagoras Theorem? That is, why can't a set of 3 prime numbers be a Pythagorean triplet?

Suppose $a$, $b$ and $c$ are three prime numbers. How to prove that $a^2 + b^2 \neq c^2$?
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### If $p$ and $q$ are prime numbers larger than $2$, then $pq + 1$ is never prime

I am trying to prove the following: If $p$ and $q$ are prime numbers larger than $2$, then $pq + 1$ is never prime. Any ideas?
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### Prove or disprive that $n^{2}-n+17$ is prime for all integers $n$

I am looking to prove this function is always prime for all integers $n$: $$n^{2}-n+17$$ I have tested it for the first $10$ integers and it seems to work but I am not sure how to prove it form all $n$...
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### The primes such that removing digits from the right end leaves another prime

The number 73,939,133 is prime. Keep removing a digit from the right end. Each of the remaining numbers is prime. How to find other numbers with this property?
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### Generalisations of primes

I've read of (normal) primes, Gaussian primes and Eisenstein primes, which all uses different ways to define an integer to be a prime. For instance, $2$ factors into $1-i$ and $1+i$ for guassian ...
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### Consequences of the negation of the Riemann hypothesis

There are many sources documenting the consequences of the Riemann hypothesis, but I can't find one discussing the consequences of its negation, particularly concerning the prime distribution. Can ...
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### Prove that if p divides xy then p divides x or p divides y

I am given that the following proposition is true. (Proved in class) "Suppose that $x$, $y\in \Bbb Z$, not both zero. Then there exists $m$, $n\in\Bbb Z$ such that $$mx + ny = d$$ where $d$ is the ...
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### Syndeticity and A.P.-richness of certain sets

Let $A \subset \mathbb{N}: \sum_{a \in A} (\frac{1}{a}) = \infty$; denote $\{ \alpha_1 @ \alpha_2: \alpha_1, \alpha_2 \in A \} = A @ A$, where "$@$" is any appropriate binary operator. (Note: $A$ is ...
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### Prime number in set $\{1,…,60\}$

How can we calculate by using the principle of inclusions and exclusions how many prime numbers are in the set $\{1, ..., 60 \}$?
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### an irreducible polynomial over GF(2) is primitive over GF(2)

let $P \in F_{2} [X]$ of degree $7$, how to prove this: P is irreducible $\Leftrightarrow$ P is primitive i tried to use the mersenne prime !
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### If $P$ and $Q$ are distinct primes, how to prove that $\sqrt{PQ}$ is irrational?

$P$ and $Q$ are two distinct prime numbers. How can I prove that $\sqrt{PQ}$ is an irrational number?
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### Disproving using a Constructive Proof

I cannot find the n to prove the negation for the following: Disprove (Prove the negation) of: For every positive integer n, $3^n + 2$ is prime The way in which I have written the negation is: ...
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### Given $0 < p < n$, prove there exists $n$ consecutive natural numbers such that each natural is divisible by at least $p$ distinct primes.

Given $0 < p < n$, prove there exists $n$ consecutive natural numbers such that each natural is divisible by at least $p$ distinct primes. Is there a general proof method to prove this ...
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### Finding all the values of n, such that $\varphi (n) = 12$ [duplicate]

I have not broken this down very far. I have come to the conclusion that there are infinitely many values for n where there exists 12 coprimes to n. Since there are infinitely many primes, and primes ...
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### Prime numbers and $\sqrt{10301}$
On my exam recently, we had the following question: Use the prime number theorem to estimate the number of primes less than $\sqrt{10301}$, and hence, give a concise argument whether 10301 is prime ...