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In Nature Vol 580, in an article about Shinichi Mochizuki's proposed proof of the abc-conjecture, there is a formulation saying:

The conjecture roughly states that if a lot of small primes divide two numbers $a$ and $b$, then only a few, large ones divide their sum, $c$.

Is that a relevant description, and if so, how to see the connection?

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    $\begingroup$ Posted on Mathematics about a month ago: Soft question about the ABC conjecture. (So far, no answers, only a few comments.) $\endgroup$
    – Martin Sleziak
    Commented May 12, 2020 at 8:00
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    $\begingroup$ This is not quite about the size of the primes, but rather about exponents in whuch they appear. If one of the numbers is divisible by a large power, that decreases its contribution to the radical. The ABC conjecture essentially states this can't happen for all of $a,b,c$ at the same time. $\endgroup$
    – Wojowu
    Commented May 12, 2020 at 8:08
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    $\begingroup$ The more the word "roughly" contributes (and it does a lot here), the less the description is relevant! $\endgroup$
    – Wolfgang
    Commented May 12, 2020 at 9:11
  • $\begingroup$ @Wojowu - In a very big and very smooth number there have to be a lot of repetitions of small primes in the decomposition. $\endgroup$
    – Lehs
    Commented May 12, 2020 at 9:28
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    $\begingroup$ @Lehs That's indeed the case, but this should be seen as a special case of ABC, not the entire content of it. For instance, when proving Fermat's Last Theorem as a corollary of ABC, we also consider numbers like $a=x^4,b=y^4,c=z^4$. These needn't be "very smooth", instead we have all prime factors appearing in exponents larger than $1$. This is enough to use ABC. $\endgroup$
    – Wojowu
    Commented May 12, 2020 at 10:09

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If you are interested in the largest prime factor of $ab(a+b)$, there is xyz conjecture.

Smooth solutions to the abc equation: the xyz Conjecture

This paper studies integer solutions to the ABC equation A+B+C=0 in which none of A, B, C has a large prime factor. Set H(A,B, C)= max(|A|,|B|,|C|) and set the smoothness S(A, B, C) to be the largest prime factor of ABC. We consider primitive solutions (gcd(A, B, C)=1) having smoothness no larger than a fixed power p of log H. Assuming the abc Conjecture we show that there are finitely many solutions if p<1.

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