If a prime $p$ divides a Fermat Number then $p=k\cdot 2^{n+2}+1$? Does anyone know a simple/elementary proof?

A little bird told me that if $2^n+1$ is prime, then $n$ is a power of $2$. I tend not to trust talking birds, so I’m trying to verify that statement independently. Suppose $n$ is not a power of $2$. Then $n = a \cdot 2^m$ for some $a$ not a power of $2$ and […]

A theorem of Édouard Lucas related to the Fermat numbers states that : Any prime divisor $p$ of $F_n=2^{2^n}+1$ is of the form $p=k\cdot 2^{n+2}+1$ whenever $n$ is greater than one. Does anyone know is there some similar theorem for generalized Fermat numbers: $F_n(a)=a^{2^n}+1$ ? I’ve been searching the internet but I couldn’t find any […]

How can I show that Fermat number $F_{5}=2^{2^5}+1$ is divisible by $641$.

Intereting Posts

Minimal polynomial of $\alpha^2$ given the minimal polynomial of $\alpha$
$y_{2n}, y_{2n+1}$ and $y_{3n}$ all converge. What can we say about the sequence $ y_n$?
Finding $\binom{999}{0}-\binom{999}{2}+\binom{999}{4}-\binom{999}{6}+\cdots +\binom{999}{996}-\binom{999}{998}$
Required reading on the Collatz Conjecture
Probability mean is less than 5 given that poisson distribution states it is 6
Can you equip every vector space with a Hilbert space structure?
What is the cardinality of the set of all non-measurable sets in $\Bbb R^n$?
How to find the vertex of a 'non-standard' parabola? $ 9x^2-24xy+16y^2-20x-15y-60=0 $
If number of conjugates of $x\in H$ in $G$ is $n$ then number of conjugates of $x$ in $H$ is $n$ or $n/2$
Let $f:\mathbb{R}\longrightarrow \mathbb{R}$ a differentiable function such that $f'(x)=0$ for all $x\in\mathbb{Q}$
How do you pronounce the symbol $'$ in $f'$?
Definition of the Brownian motion
Why Do Structured Sets Often Get Referred to Only by the Set?
Convergence in probability inverse of random variable
Problem with an algorithm to $3$-colour the edges of cubic graphs