Linked Questions

5
votes
2answers
974 views

Prove that $\gcd(2^{2^m}+1,2^{2^n}+1)=1$ if $m,n$ are positive integers. [duplicate]

Prove that $\gcd(2^{2^m}+1,2^{2^n}+1)=1$ if $m,n$ are positive integers. Let $d=\gcd(2^{2^m}+1,2^{2^n}+1)$, then $d\mid 2^{2^m}+1$ and $d\mid2^{2^n}+1$ and then $d\mid2^{2^m}+1-2^{2^n}-1$, i.e. $d\...
1
vote
2answers
199 views

Fermat Numbers Proof [duplicate]

Fermat numbers are shown by: $F_m = 2^{2^m} + 1$. How can I prove that for any $m ≠ n$, I can have $(F_m, F_n) = 1$?
1
vote
2answers
202 views

A Number Theory problem (GCD) [duplicate]

Prove that $$gcd(2^{2^m}+1, 2^{2^n}+1)=1$$ if $m, n$ are positive integers such that $m \neq n$. A Hints to solve the problem is also given in the book as follows: Let $m>n$. Then $2^m=2^{n}2^{m-...
-1
votes
2answers
85 views

Prove that $\gcd(g_a,g_b) = 1$ given that for $n \in Z^{\geq 0}$, define $g_n = 2^{2^n} + 1$ [duplicate]

Prove that $\gcd(g_a,g_b) = 1$ given that for $n \in Z^{\geq 0}$, define $g_n = 2^{2^n} + 1$. I have already proved that $g_0\cdot g_1\cdots g_{n-1} = g_n -2$ if this hint is useful in this proof.
0
votes
1answer
92 views

$(a^{2^n}+1,a^{2^m}+1)=1 or 2$ [duplicate]

Prove that if $m\not =n,a$ are positive integers then $(a^{2^n}+1,a^{2^m}+1)$ is $1$ if $a$ is even and $2$ if $a$ is odd. I solve the problem in the following way: I assume that $m>n$ then $$a^{...
0
votes
1answer
65 views

Problems regarding GCD [duplicate]

If $m > n$ and $a,m,n$ are positive, with $m$ not equal to $n$, find the greatest common divisor of $2^{2^m}+1, 2^{2^n}+1$. Please solve this problem using Euclid's algorithm. I tried to use ...
0
votes
0answers
47 views

Fermat Prime Numbers Coprime [duplicate]

Fermat numbers are shown by: $F_m = 2^{2^m} + 1$. How can I prove that for any $m ≠ n$, I can have $(F_m, F_n) = 1$, or coprime?
2
votes
3answers
4k views

Prove that distinct Fermat Numbers are relatively prime [duplicate]

The Fermat numbers are defined by $F_m = 2^{2^m} + 1$. Prove that for $m \ne n$ we have $(F_m, F_n) = 1$. I have to first prove that $F_{m+1} = F_0F_1 \cdots F_m + 2$ by representing $F_{m+1}$ in ...
2
votes
1answer
187 views

Show that Fermat number $F_n$ and its index $n$ are coprime.

I want to show that $\gcd(F_n,n)=1$, where $F_n=2^{2^n}+1$. How to prove this? I can show that that $\gcd(F_n, F_m)=1$ for any natural $n$ and $m$, and that $F_{n+1}=(F_n)^2-2F_n+2=F_0\dots F_{n-1}+2$...
1
vote
1answer
300 views

Co prime numbers

How to find a set of numbers which are coprime to each other (all numbers are pairwise co prime) ? the numbers can be assumed to be less than a specific integers. like all numbers in the range [0,x]
1
vote
0answers
171 views

Proving that Fermat Numbers are coprime

Here's a problem on one of the past pages that was incompletely settled, i arrived at a contradiction, and i don't know if someone can show otherwise; Assume that if $\phi_n=2^{2^n}+1$ then g.c.d($\...
0
votes
1answer
79 views

Finding the HCF

Find $(a^{2^m}+1, a^{2^n}+1)$ when a is odd and a,m,n are positive integers and m is not equal to n. I know that the hcf is a multiple of two but I can't prove that it is 2 which is the answer. Plz ...
0
votes
1answer
59 views

If $m\neq n$ what is $\mathrm{gcd}(a^{2n}+1,a^{2m}+1)?$ [closed]

If $m\neq n,$ compute $\mathrm{gcd}(a^{2n}+1,a^{2m}+1).$ In my question, $m$ , $n$ , and $a$ are positive integers.
0
votes
2answers
53 views

Prove that $\frac{2^{2^n}+1}{2^{2^m}+1}$ is irreducible, if $n>m\geq 0$ are integers.

Prove that $\frac{2^{2^n}+1}{2^{2^m}+1}$ is irreducible, if $n>m\geq 0$ are integers. I want to show by induction that $gcd(a+1,a^{2k}+1) = gcd(a+1,2)$ and i have to choose $a$. I choose $a=2^{2^...
0
votes
1answer
38 views

Show that the only divisors are $1$ and $2$ for $(z^{(2^x)}+1)$ and $(z^{(2^y)}+1)$ where $x,y,z\in\mathbb{N}$

I am trying to show that the only divisors are $1$ and $2$ for both $(z^{(2^x)}+1)$ and $(z^{(2^y)}+1)$ where $x,y,z\in\mathbb{N}$. To start the problem, the logical choice is to use difference of ...

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