The set $A$ = {$X_n\mid n\in \Bbb N$} where $X_n = a^{n+1} + a^{n} - 1$, with $a \gt 1, a \in \Bbb Z$.
Show that there are infinitely many numbers that are pairwise coprime.
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$(a^{n+2}+a^{n+1}-1,a^{n+1}+a^n-1)$ $=(a^{n+2}-a^{n},a^{n+1}+a^n-1)$ $=((a^{2}-1)a^{n},a^{n+1}+a^n-1)$ $=(a^{2}-1,a^{n+1}+a^n-1)$ as $(a^n,a^{n+1}+a^n-1)=1$ $=((a-1)(a+1),a^n(a+1)-1)$ $=(a-1,a^n(a+1)-1)$ as $(a+1,a^n(a+1)-1)=1$ $=(a-1,a^n(a-1)+2(a^n-1)+1)=1$ |
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Assume $n>m$. Let $d=n-m$. Since integer linear combinations of multiples of $\gcd(X_n,X_m)$ are again multiples of $\gcd(X_n,X_m)$, we find that $X_n-a^dX_m = a^d-1$ is a multiples of $\gcd(X_n,X_m)$. Consider the case that $n$ is a proper multiple of $m$. Then $d$ is also a multiple of $m$ and using the well-known fact that $x-y$ divides $x^k-y^k$, we see that $a^m-1$ is a multiple of $a^d-1$, hence $X_m - (a+1)(a^m-1)= a$ is a multiples of $\gcd(X_n,X_m)$. Finally, since $d>0$ we have that $a^{d-1}\cdot a-(a^d-1)=1$ is a multiple of $\gcd(X_n,X_m)$, i.e. $\gcd(X_n,X_m) = 1$. Thus all we have to do is take an infinite subset $M\subseteq \mathbb N$ such that any tow elements of $M$ are multiples of each other. For example, one could take all powers of $2$ as $M$. Then $B:=\{X_m\mid m\in M\}$ has the required properties: It is infinite (because $a>1$) and any two distinct elements of $B$ are coprime. Remark: We used $a>1$ only to conclude that $B$ is infinite! |
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