[7430,3459,2227]

Hi @Peter! I found an interesting question on MSE

I’m wondering if you can help

It is a generalisation of an unanswered question on MSE. I think that we have only one solution to $$a+b=\gcd(a,b)^n$$, $$b+c=\gcd(b,c)^n$$, $$c+a=\gcd(c,a)^n$$ where $a,b,c,n\in\mathbb N$ which is $(a,b,c,n)=(2,2,2,2)$

For $n=1$ it is obvious there are no solutions. For $n\geq 3$, we can show that $\gcd(a,b,c)=1$. Then by adding all equations $\bmod 2$ we also have that $a,b$ must be even and $c$ odd WLOG.

Even for $n=2$ we have that $\gcd(a,b,c)$ must be a power of $2$.

Did you check a reasonable range , say a,b,c <= 100 ? If there is no further solution , we can begin to think about a proof strategy.

the case for $n=3$ was checked till 200,000

without any solution

ive checked for $n=2$ and only found $(2,2,2)$

That is a huge range. Which partial results could you prove ? The case $n=2$ ?

I could only prove the points mentioned earlier IE For $n\geq 3$, $gcd(a,b,c)=1$. For $n=2$, $\gcd(a,b,c)$ is a power of $2$. For all $n$, it’s also easy to show that $\gcd\left(b,\frac{a}{\gcd(a,b)}\right)=1$ and $\gcd\left(a,\frac{b}{\gcd(a,b)}\right)=1$ and similarly for $b,c$, and $c,a$. but I’m not sure if that is helpful

The problem seems intereting , but at the moment I have no clue how to solve it.

it might not be easily prove-able. We might need some numerical results to verify it etc.

this is the special case of this question asked on the site

it has many upvotes, many views but no conclusive answer

We have $gcd(a^3,b^3)=gcd(a,b)^3$ , right ? So this is the case $n=3$.

Yes.

FYI, I found a thread for $n=2$ using approach0 search just now

on AoPS

artofproblemsolving.com/community/c6h2072577p14849026

I currently check a,b,c <= 10^4 with arbitary n. No solution yet found.

Wow. That was fast

I am not finished yet.

a=1862 (WLOG a<=b)

Oh,right

Can infinite descent help ?

I think it can be tried. I didn't try much on that

I will work on this problem. If I found something interesting , I will post it here.

Alright. Thanks for your help as always

@Peter This is one of your best questions! Thumbs up!

You have also shown why it is so important to find a factor in this case.

And then Izaak van Dongen gave us an answer!

For me, this is a prime example of applied maths. A goal can only be achieved if you work together as a team.

Hello