
> 
> Could You give your ideas, your comment, or a referen for a conjecture as follows: 
> 
> 
> 


Consider $A, B, C$ be three positive integers numbers. By [Fundamental theorem of arithmetic](https://en.wikipedia.org/wiki/Fundamental_theorem_of_arithmetic) we write:


$A=a\_1^{x\_1}a\_2^{x\_2}...a\_n^{x\_n}$, 


$B=b\_1^{y\_1}b\_2^{y\_2}...b\_m^{y\_m}$, 


$C=c\_1^{z\_1}c\_2^{z\_2}...c\_k^{z\_k}$


Let $N=\min\{x\_i, y\_j, z\_h \}$ where $1 \le i \le n, 1\ \le j \le m, 1\le h \le k$



> 
> For a positive integer $N\_0 > 3$, there exist only finitely many triples $(A, B, C)$ of coprime positive integers, with $A+B=C$, such that: $N \ge N\_0$
> 
> 
> 


**Edited:** I repalced $N\_0 \ge 3$ by $N\_0 > 3$



Another conjecture on sum $A+B=C$
This conjecture is false, though only barely. 


It should be true for $N\_0 \geq 4$ by the ABC conjecture.
But for $N\_0 = 3$ there are infinitely many counterexamples such as
$$
191114642^3 + 4309182809^3 = 7^4 \, 321817873^3.
$$
I obtained this by starting from the point 
$P\_1 = (2:1:1)$ on the elliptic curve $x^3+y^3=7z^3$ 
(with origin $P\_0 = (1:-1:0)$), and looking for a multiple 
$P\_n = nP\_1$ whose $z$-coordinate is divisible by $7$; such $n$ must exist because $7|z(P\_n)$ **iff** $P\_n$ reduces mod $7$ to $P\_0$,
and here it turned out that $n=7$ was the first example.
Then any $n=7m$ works, so we get an infinite sequence of counterexamples.


The same construction works for any elliptic curve $x^3+y^3=Dz^3$
of positive rank.


P.S. I see that a simpler counterexample$$
271^3 + 3^5 146^3 = 919^3
$$
was given by **user459832** to 
[the parallel question
on Math Stackexchange](https://math.stackexchange.com/questions/2345108/another-conjecture-on-sum-ab-c). This corresponds to the third multiple of
the generator $(2:1:1)$ on $x^3+y^3=9z^3$; here the first case of $3|z$
is small enough that we can still comfortably exhibit the next one, 
$$
415280564497^3 + 676702467503^3 = 3^5 116223894220^3.
$$


