The Borel-Cantelli lemma is very frequently used to give a heuristic for whether or not certain statements in number theory are true.

For example, it gives some evidence that there are finitely many Fermat Primes, that is primes of the form $F_n=2^{2^{n}}+1$. Since the asymptotic density of the primes around $x$ is $\frac{1}{\log x}$, we expect that $$\text{Pr}\left(F_n\text{ is a prime number}\right)\approx \frac{1}{2^n}.$$ As the series $\sum_{n=1}^\infty \frac{1}{2^n}$ converges, Borel-Cantelli suggests that there will be finitely many Fermat Primes.

**More examples:**

- Infinitude of the Mersenne Primes, since $\sum_{n=1}^\infty \frac{1}{n}$ diverges.
- Heuristic justification for the ABC conjecture on Tao's blog.
- Two different heuristics that there are only finitely many elliptic curves of rank greater than $21$. The first is due to Granville, and the second is recent work of of Garton, Park, Poonen, Voight and Wood.

In the case of Mersenne primes and Fermat primes, some major assumptions are made about independence, but even then, Borel-Cantelli can only ever be a heuristic since a "probability $0$ event" could still occur.

My question is:How reliable is this heuristic? Are there any known cases/past conjectures where the Borel-Cantelli heuristic has been incorrect?

**Edit:** As Terence Tao mentions, the Borel-Cantelli heuristic fails for the $n=3$ case of Fermat's last theorem due to algebraic structure. Examples like this are exactly what I am looking for.

not local: if $f(0) \not= 0$ then $\mu(f^8+u^3) = 1$, so $f^8 + u^3$ can't be irreducible (it's easier to see this if $f(0) = 0$). The calculation of the M\"obius function of $f^8+u^3$ is, as you say, not obvious. $\endgroup$ – KConrad Apr 16 '15 at 4:512more comments