Intereting Posts

Cantor construction is continuous
Does the likelihood of an event increase with the number of times it does not occur?
How do I find the period of repetition of this modular equation?
Sum over all contiguous partitions of an array
The square of minimum area with three vertices on a parabola
Prove that $|\sin z| \geq |\sin x|$ and $|\cos z| \geq |\cos x|$
Does a plane curve with polar equation $r=\lambda_1\cos^2\theta+\lambda_2\sin^2\theta$ have a name?
Is the localization of a PID a PID?
How to prove that $ = $ given $H \le G$?
Arithmetic on $$: is $0 \cdot \infty = 0$ the only reasonable choice?
The geometric interpretation for extension of ideals?
Show that, for all $n > 1: \frac{1}{n + 1} < \log(1 + \frac1n) < \frac1n.$
$Tf=\sum\limits_{n=1}^\infty f(n)x_n$ is surjective from $\ell^1$ to a separable Banach space
How do the $L^p$ spaces lie in each other?
The adjoint of left exterior multiplication by $\xi$ for hodge star operator

So today in my final for number theory I had to prove that the Fermat numbers ($F_n=2^{2^n}+1$) are coprime.

I know that the standard proof uses the following: $F_n=F_1\cdots F_{n-1}+2$ and then the $\gcd$ divides $2$, and it cannot be two, and hence the numbers are coprime.

However, he asked us to use the hint: “Let $l$ be a prime dividing $F_n$. What can you say about the order of $2$ in $(\mathbb{Z}/l\mathbb{Z})^\times$?”

- parametric solution for the sum of three square
- the number of the positive integer numbers $k$ that makes for the two quadratic equations $ \pm x^2 \pm px \pm k$ rational roots.
- Number Theory Prime Reciprocals never an integer
- Why is $n_1 \sqrt{2} +n_2 \sqrt{3} + n_3 \sqrt{5} + n_4 \sqrt{7} $ never zero?
- Sequence of positive integers such that their reciprocals are in arithmetic progression
- Successive ratios of a sequence, is this limit true?

I have been thinking about it and I cannot figure out how to use his hint.

Any ideas?

- Show that $u_1^3+u_2^3+\cdots+u_n^3$ is a multiple of $u_1+u_2+\cdots+u_n$
- Relationships between the elements $(a,b,c,d)$ of a solution to $A^2+B^2+4=C^2+D^2$
- Evaluate a character sum $\sum\limits_{r = 1}^{(p - 1)/2}r \left( \frac{r}{p} \right) = 0$ for a prime $p \equiv 7 \pmod 8$
- Is there a rational surjection $\Bbb N\to\Bbb Q$?
- There is a number of three digits, 1 is never immediate right of 2 is?
- A question about an asymptotic formula
- Sum of odd prime and odd semiprime as sum of two odd primes?
- Solutions of $p!q! = r!$
- Knight move variant: Can it move from $A$ to $B$
- Books about the Riemann Hypothesis

If $l$ is a prime that divides $F_n$, then $2^{2^n}\equiv -1 \pmod{l}$, and therefore $2^{2^{n+1}}\equiv 1 \pmod{l}$. So $2$ has order $2^{n+1}$ modulo $l$. Equivalently, but in more group-theoretic language, (the equivalence class of) $2$ has order $2^{n+1}$ in $(\mathbb{Z}/l\mathbb{Z})^\times$. (The order of $2$ must divide $2^{n+1}$, but cannot be $\le 2^n$.)

If $p$ is a prime that divides $2^{2^k}+1$, then by the same reasoning $2$ has order $2^{k+1}$ modulo $p$. If $k<n$, that means that $p$ cannot be equal to $l$. So we have proved that no prime that divides $F_n$ can divide any $F_k$ with $k<n$. This shows that $F_n$ and $F_k$ are relatively prime.

**Remark:** We do not really need to identify the order of $2$ explicitly. For if $l$ is a prime that divides $F_n$, then $2^{2^n}\equiv -1 \pmod{l}$. But if $p$ is a prime that divides $F_k$ for some $k<n$, then $2^{2^k}\equiv -1\pmod{p}$, and therefore $2^{2^{k+1}}\equiv 1\pmod{p}$. Since $k+1 \le n$, this is impossible.

Let $A_r=a^{2^r}+1,$

$A_{n+t}=a^{2^{n+t}}+1=(a^{2^n})^{2^t}+1=(A_n-1)^{2^t}+1\equiv2\pmod{A_n}$ for integer $t\ge1$

$\implies(A_{n+t},A_n)=(2,A_n)$

If $a$ is even, $A_r$ will be odd if $r\ge1$

$\implies(2,A_n)=1$ for $n\ge1$

- How to calculate the payout rate of a slot machine?
- Surface integral over ellipsoid
- Use mathematical induction to prove that any integer $n\ge2$ is either a prime or a product of primes.
- Limit of $\sum_{i=1}^n \left(\frac{{n \choose i}}{2^{in}}\sum_{j=0}^i {i \choose j}^{n+1}\right)$
- $f$ continuous iff $\operatorname{graph}(f)$ is compact
- Is $\sum_{k=1}^{n} \sin(k^2)$ bounded by a constant $M$?
- Favorite Math Competition Problems
- Why $ \sum_{k=0}^{\infty} q^k $ sum is $ \frac{1}{1-q}$ when $|q| < 1$
- Can Path Connectedness be Defined without Using the Unit Interval?
- What comes after tetration ? And after ? And after ? etc.
- fundamental period
- Description of the Universe $V$
- Intuition: If $a\leq b+\epsilon$ for all $\epsilon>0$ then $a\leq b$?
- Ring of $p$-adic integers $\mathbb Z_p$
- Can every real number be represented by a (possibly infinite) decimal?