An article online says:
The abc conjecture refers to numerical expressions of the type $a + b = c$. The statement, which comes in several slightly different versions, concerns the prime numbers that divide each of the quantities $a$, $b$ and $c$. Every whole number, or integer, can be expressed in an essentially unique way as a product of prime numbers-- those that cannot be further factored out into smaller whole numbers: for example, $15 = 3 \times 5$ or $84 = 2 \times 2 \times 3 \times 7$. In principle, the prime factors of a and b have no connection to those of their sum, c. But the abc conjecture links them together. It presumes, roughly, that if a lot of small primes divide a and b then only a few, large ones divide c.
My question related to this sentence therein:
In principle, the prime factors of $a$ and $b$ have no connection to those of their sum, $c$.
I don't understand that statement, since $$c = (\text{the product of the prime factors of }a) + (\text{the product of the prime factors of }b).$$
That certainly seems like a connection!
I do not have strong math chops, and would appreciate an answer in English rather than in notation, to the extent possible. I fully realize I'm just not grasping what's being said, I'm certainly not critiquing the conjecture.
You seem to assume that $a$ equals the product of its prime factors (and likewise for $b$), but that doesn't take into account multiplicity of factors. For example, the only prime factors of 72 are 2 and 3, but 72 is not $2\times 3$.
Even if you could get $c$ from just the prime factors of $a$ and $b$, that wouldn't tell you much about the prime factors of $c$ (and especially about their sizes) unless you went through the work of factoring $c$.