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It is well known that the ABC conjecture gives an immediate proof of Fermat Last Theorem (FLT). It seems that it proves something stronger involving not necessarily perfect power, which may still be open.

We first observe there is a natural extension of the notion of a perfect power of an integer to all positive integers. We define the average power of a positive integer $m$ to be $ap(m):=\frac{\log(m)}{\log(rad(m))}$, where $rad(m)$ is the product of primes dividing $m$. For positive integers $m,k$, we have $ap(m^k)=k.ap(m) \ge k$ with equality iff $m$ is squarefree. $ap(3^2.5.7)>1$ but is less than $ap(3.5.7^2)$. In fact every positive integer $m$ has a unique representation as $m=m_1m_2^2m_3^3...$ where $m_j$ is the product of primes $p$ such that $p^j$ exactly divides $m$ so that $ap(m)=\frac{\sum j \log(m_j)}{\sum \log(m_j)}$ is indeed the correctly weighted average of individual powers in $m$.

We shall say that a triple $(a,b,c)$ of positive integers is a abc tuple if $c=a+b$ and $gcd(a,b)=1$. The ABC conjecture states that for every $\epsilon >0$, and any abc tuples with at most finitely many exceptions, $c <rad(abc)^{1+\epsilon}$. This implies $abc \le 4abc \le c^3 < rad(abc)^{3+3\epsilon}$. So that we have $ap(abc)<3+\epsilon$ for every abc tuple with at most finitely many exception. It follows that the condition $$minap:=min\{ap(a),ap(b),ap(c)\} \ge 3+\epsilon,$$ which clearly implies $ap(abc) \ge 3+ \epsilon$, can only holds for at most finitely many abc tuples.

An effective version(no exception) of ABC for $\epsilon=1$ would imply that there is no abc tuple with $a,b,c$ each of average power $\ge 4$, and this is an enhanced version of FLT since it applies to all abc tuples, not just tuple of perfect powers. One natural question is does this enhanced version of FLT follow from the extension of Taylor and Wiles' approach? It can be phrased as an additive problem : that the set $\{m: ap(m) \ge 4\}$ is sum free and apparently cannot be phrased in terms of finitely many Diophantine equations.

The known abc tuple $$2^5.11^2.19^9+5^{15}.37^2.47=3^7.7^{11}.743, ap(abc)=4.2680083..., minap=3.69810313...$$ suggests that $4$ may be the correct power.

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    $\begingroup$ I don't think I've seen your average power of an integer as a function explicitly stated before and it seems like an interesting function. The idea of it seems to be implicitly in some of the literature on the ABC conjecture. For example, Luca and Pomerance's "On the radical of a perfect number" uses bounds on the radical of a perfect number $N$ in terms of a power of $N$ to show that ABC implies that for any given $k$ there are only finitely many perfect numbers $x$ and $y$ where $x-y=k$. $\endgroup$
    – JoshuaZ
    Sep 10, 2021 at 11:29
  • $\begingroup$ Thanks for your reference. It seems that $ap(abc)$ has been considered in connection with Szpiro conjecture. $abc$ tuples with the largest known values of $ap(abc)$ are tabulated in nitaj.users.lmno.cnrs.fr/abc.html and are called $abc$-Szpiro examples, though it does not seem to have been mentioned that they measure average power. It seems to be a more natural quantity to consider and it allows us to state a strengthened version of FLT. $\endgroup$
    – CHUAKS
    Sep 11, 2021 at 21:45
  • $\begingroup$ Another indication that ap is a useful term to work with is the inequality. If $m_j$ are pairwise relatively prime positive integers, we have $min\{ap(m_1),..,ap(m_n)\} \le ap(m_1...m_n) \le max\{ ap(m_1),...,ap(m_n)\},$ so ap turn multiplication to something which behave like an additive average. $\endgroup$
    – CHUAKS
    Sep 27, 2021 at 4:35
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    $\begingroup$ ap(m) has also been defined and called power function in ams.org/notices/200002/fea-mazur.pdf. It seem strange this paper was not widely quote $\endgroup$
    – CHUAKS
    Jan 30, 2022 at 23:10

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