Prove that $\gcd\left(n^{a}+1, n^{b}+1\right)\mid (n^{\gcd(a, b)}+1)$

Question

Let $a$ and $b$ be positive integers. Prove that $\operatorname{gcd}\left(n^{a}+1, n^{b}+1\right)$ divides $n^{\operatorname{gcd}(a, b)}+1$.

My work -

I proved this for $n=2$ but I am not able to prove this for all $n$ (if anyone wants I can give my proof for $n=2$).

More Observation.

If $a$ and $b$ are both odd, then $d=\gcd(a,b)$ is an odd positive integer. Therefore, $$n^a+1=(n^d+1)\left(n^{d(a-1)}-n^{d(a-2)}+\ldots-n^d+1\right)$$ and $$n^b+1=(n^d+1)\left(n^{d(b-1)}-n^{d(b-2)}+\ldots-n^d+1\right),$$ whence $n^d+1$ divides both $n^a+1$ and $n^b+1$. That is, $n^d+1$ divides $\gcd(n^a+1,n^b+1)$. However, we can perform Euclidean algorithm as follows.

Without loss of generality, let $a\geq b$.

Case I: $a\geq 2b$. We have $$n^a+1=(n^{b}+1)\left(n^{a-b}-n^{a-2b}\right)+(n^{a-2b}+1)\,.$$ We can replace $(a,b)$ by $(a-2b,b)$, and perform more reduction steps.

Case II: $b<a<2b$. We have $$n^{a}+1=(n^b+1)n^{a-b}-\left(n^{a-b}-1\right)$$ and $$n^b+1=\left(n^{a-b}-1\right)n^{2b-a}+(n^{2b-a}+1)\,.$$ Thus, we can replace $(a,b)$ by $(b,2b-a)$ and perform more reduction steps.

Case III: $a=b$. Then, the reduction steps end.

Note that, at each step, the difference between $a$ and $b$ never increases. (Observe that, we cannot perform the steps in Case II infinitely many times, as the smaller value between $a$ and $b$ always decreases.) Therefore, the process has to stop when both numbers become the same odd integer $s$, which is an integer combination of $a$ and $b$. However, $d$ divides any integer combination of (the starting values of) $a$ and $b$. Thus, $d$ divides $s$. The Euclidean algorithm above shows that $n^s+1$ is the greatest common divisor of $n^a+1$ and $n^b+1$. Thus, $s=d$, so in the case $a$ and $b$ are odd, $$\gcd(n^a+1,n^b+1)=n^{\gcd(a,b)}+1\,.$$

Answer 1

Let $\mathrm{WLOG}$ $a>b$. For any prime $p$ let $v_p(m)$ denotes the maximum exponent of $p$ in the canonical prime factorisation of $m$. We need to show that $$v_p(\mathrm{gcd}(n^a+1,n^b+1))\leq v_p(n^{\mathrm{gcd}(a,b)}+1)$$ For all primes $p$. If $v_p(n^{\mathrm{gcd}(a,b)}+1)=0$, then it's your exercise why $p$ doesn't divide $\mathrm{gcd}(n^a+1,n^b+1)$. Now let $$v_p(\mathrm{gcd}(n^a+1,n^b+1))=\alpha\,.$$ Then $p^{\alpha}\mid (n^a+1)$ and $p^{\alpha}\mid(n^b+1)$. Therefore, $$p^{\alpha}\mid n^a-n^b= n^b(n^{a-b}-1)\,.$$ Since $p>1$, $\mathrm{gcd}(n,p)=1$. Then, $p^{\alpha}\mid (n^{a-b}-1)$. Similarly we get, $$p^{\alpha}\mid (n^{a-b}-1)+(n^b+1)=n^b(n^{a-2b}+1)\,.$$

Then as before, $p^{\alpha}\mid(n^{a-2b}+1)$.

In this way you can reach $\mathrm{gcd}(a,b)$ in the exponent like we get gcd of two integers by Euclidean algorithm.

Hence finally you will conclude that $p^{\alpha}\mid (n^{\mathrm{gcd}(a,b)}+1)$. Hence $v_p(n^{\mathrm{gcd}(a,b)}+1)\geq \alpha$.

Done!

Answer 2

Suppose that for some prime $p$ and positive integer $k$ we have $p^k$ divides both $n^a+1$ and $n^b+1$. Then, we need to prove that $p^k$ divides $n^{\gcd(a,b)}+1$. Denote $d=\gcd(a,b)$. Here, we will consider two cases:

Case 1. $p=2$. In this case, if $a$ or $b$ is even, then $k=1$ (because $m^2+1$ can't be divisible by 4) and $n$ should be odd. So, $n^d-1$ is divisible by $p^k=2$, as desired.

If both $a$ and $b$ is odd, then $\gcd(n^a+1, n^b+1)=n^d+1$ (it's similar to Prove that $\gcd(a^n - 1, a^m - 1) = a^{\gcd(n, m)} - 1$) and in particular, $2^k\mid n^d+1$.

Case 2. $p>2$. In this case, note that $p^k$ divides $$n^{2a}-1=(n^a-1)(n^a+1)$$ and $$n^{2b}-1=(n^b-1)(n^b+1)\,,$$ so $p^k$ divides $n^{2d}-1=(n^d-1)(n^d+1)$. Note that $p$ can't divide both $n^d-1$ and $n^d+1$ (because $p>2$). Hence, it's sufficient to prove that $n^d-1$ can't be divisible by $p^k$. Indeed, if $n^d\equiv 1\pmod {p^k}$, then $$n^a\equiv n^b\equiv 1\pmod {p^k}\,.$$ However, by our assumption we have $n^a\equiv n^b\equiv -1\pmod {p^k}$, so due to $p^k>2$ we get a contradiction. Thus, $p^k$ divides $n^d+1$ as desired.