# 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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### Do all primes occur in some sequence associated with the Collatz conjecture?

Let $f(n) = \begin{cases} n/2, & \text{if$n$is even} \\ 3n+1, & \text{if$n$is odd} \end{cases}$ For an arbitrary prime $p$ are there some start value $x_0$ such that $p = x_k$ for some ...
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### Show that a positive integer $a>1$ is a perfect square (i.e., the square of an integer)…

if and only is in the prime decomposition of $a$ all the exponent are even integers. I don't understand what the question is asking. If I'm interpreting this correctly....any $a>1$ such as 9 would ...
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### Generating all prime powers $\leq N$

Some very good algorithms exist to generate all primes $p$ up to some bound $N$, like the sieve of Erastothenes and the sieve of Atkin. However, suppose I want to generate a (sorted) list of all prime ...
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### Field of the form $\{a+bi|a,b\in \mathbb{F}_p\}$

Artin, Algebra, Chapter 3, Ex 1.11 Consider whether the set of symbols $\{a+bi|a,b\in\mathbb{F}_p\}$ forms a field, if the laws of composition are made to mimic addition and multiplication of complex ...
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### Prove amount of primes of form 4n-1 is infinite, looking for explanation of last part

This is an exercise in Bigg's Discrete Mathematics (Oxford Press). It is stated roughly like this: Suppose that there are finitely many primes of this form $(4n - 1): 3, 7, 11, 19,...,X$. ...
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### Algebraic number fields in which all rational primes are inert

Is there an algebraic number field $F\supsetneq\mathbb{Q}$ such that all rational primes are inert in $\mathcal{O}_F$?
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### The least prime factor of this super gigantic number (sequence)

I am planning to submit this sequence to oeis, and it is: $a(n)$ is the least prime divisor of $$2^{3^{5^{7^{11^{...^{p(n-1)^{p(n)}}}}}}}+p(n)^{p(n-1)^{...^{11^{7^{5^{3^{2}}}}}}}$$ Where the power ...
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### How Euclidian Algorithm for division works with algebric expressions?

I am attending an introductory Number Theory class for Computer Science focused on cryptography. I have done some exercises with integers number but I have two exercises in which appears algebric ...
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### Weird Prime Number Discovery [closed]

I noticed a strange phenomenon while examining prime numbers. Here it is: We'll say num = number If num's sum of digits is 4 and num is not even, num is prime. Can somebody explain to me why this ...
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### If for $p \in \Bbb P$ and $x,y,z \in \Bbb N$ we have $x^{p-1}+y^{p-1}=z^{p-1}$, then $p\mid xyz$

I want to prove the statement in the title. This is, how far i came: Proof. We have $p \in \Bbb P$ and $x,y,z \in \Bbb N$ with $x^{p-1}+y^{p-1}=z^{p-1}$. If $p=2$, we have $x+y=z$. Now if $x$ and $y$...
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### Number of solutions to a modular equation of a specific form

I struggle with this Exercise, or at least the part where one should prove how many solutions there are. Simply inserting f=0 contradicts the suggested number of solutions. Let $p$ be an odd prime,...