Tagged Questions

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Primality of $F_{q,j}=(2^{q^{j+1}}-1)/(2^{q^j}-1)$ , when $q$ is prime, $j\ge0$?

Let $P_{q,j}=(2^{q^{j+1}}-1)/(2^{q^j}-1)$ , $q$ prime and $j\ge0$. $P_{2,j}$ is a Fermat number, $P_{q,0}$ is a Mersenne number. Apart from Fermat primes and Mersenne primes, and apart from ...
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Why is Euler's Totient function always even?

I want to prove why $\phi(n)$ is even for $n>3$. So far I am attempting to split this into 2 cases. Case 1: $n$ is a power of $2$. Hence $n=2^k$. So $\phi(n)=2^k-2^{k-1}$. Clearly that will ...
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determining order of $x+1$ given the $x$ has order three

I was trying to expand $(x+1)^n$, then plug $x^3$ in to the expansion of the $(x+1)^n$, keep trying it until I get the order, are there any other ways? So if $x^3\equiv 1\pmod y$, how would I ...
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observation on poulet numbers need generalizations

Let us take two different poulet numbers with common prime factor. For example, CASE-1: $p_1$ $= 341 = (11)(31)$ and $p_2$ $= 4681 = (151)(31)$, as $31$ is common prime factor. let us define some ...
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Find all $n\in\mathbb N$, $n>3$ such that $p(n)=2n+16$, where $p(n)$ is the product of all the prime numbers less than $n$.

Find all $n\in\mathbb N$, $n>3$ such that $p(n)=2n+16$, where $p(n)$ is the product of all the prime numbers less than $n$. E.g. $p(7)=2\cdot3\cdot5$ (and $n=7$ is a solution). Let ...
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Cyclic rearrangements of periods of certain periodic numbers

A student of mine observed the following \begin{align} \frac{1}{7}=0.\overline{142857} &\qquad \frac{2}{7}=0.\overline{285714} &\qquad \frac{3}{7}=0.\overline{428571} \\ ...
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In a given sequence of consecutive integers, finding the count of integers with a least prime factor greater than $p$

If a number $x$ has a least prime factor of $3$, then it is necessarily of the form $6y+3$ and the next number with a least prime factor of $3$ is $6y+9$. Between these two numbers there are always ...
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Generating of primes in base-3 edited

How to prove the following statement! for example primes $p_1$ = $7$ = $n$ and $p_2$ = $13$ = $2n-1$(each prime is $> 3$), then $m = p_1 p_2$ is a Fermat-pseudo prime in base-3. Can we prove ...
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Fermat Primes re edited [duplicate]

First of all sorry for sending the same question. Is my cited below observations are true? If yes, how to prove? 1) Many of $poulet$ numbers are in the form of $(4^x -1)$/$3$, where $x$ is some ...
Let rad(n) = $\Pi_{primes, p|n}$ p.
Let $\operatorname{rad}(n) = \displaystyle\prod_{\stackrel{p|n}{p \text{ prime }}}p$ . I have proven that $\operatorname{rad}(n)$ is a multiplicative arithmetic function. I have also proven that ...