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.
Best Answer
There is an example of Elkies in Appendix II (page 718, and 13 in the PDF) of this paper.
The context therein is that we want "large" integral points on elliptic curves, so $y^2=x^3+ax+b$ with each of these an integral parameter. The proper yardstick by which to measure is to try to maximize $x$ in terms of $a$ and $b$, namely by the quotient $\rho=\frac{\log x}{\log \max(|a|^{1/2},|b|^{1/3})}$. Considering $|a|\sim T^2$ and $|b|\sim T^3$, the probabilistic heuristic says $x^3+ax+b$ is square about $1/x^{3/2}$ of the time (for large $x\gg T$), and summing gives $$\sum_{x\sim T^\lambda}\sum_{a\sim T^2}\sum_{b\sim T^3} \frac{1}{x^{3/2}}\approx \frac{T^5}{T^{\lambda/2}}$$ for the number of such integral points in a dyadic $x$-interval of size $T^\lambda$. So when $\lambda \ge 10+\epsilon$, there should be no solutions as $T\rightarrow\infty$ (equivalently stated, for every $\epsilon \gt 0$ there should be finitely many integral points with $\rho\ge 10+\epsilon$).
But Elkies cleverly recalls Pell, and considers the polynomial equation $$Q(t)Y(t)^2=X(t)^3+A(t)X(t)+B(t)$$ for polynomials $A,B,Q,X,Y$ with $Q$ quadratic. The idea will then be to specialize the $t$-parameter so that $Q(t)$ is square (if there is one integral $t$ with $Q(t)$ square, there are infinitely many). By counting parameters he finds there are four cases (of degrees of $A,B,X,Y$) with prospects of a solution with $\rho\rightarrow 12$ as $t\rightarrow\infty$, and in the smallest case this solution is defined over the rationals: $$X(t)=6(108t^4-120t^3+72t^2-28t+5)$$ $$Y(t)=72(54t^5-60t^4+45t^3-21t^2+6t-1)$$ $$Q(t)=2(9t^2-10t+3),A(t)=132,B(t)=-144(8t-1)$$ with $X(t)\sim B(t)^4/2^{25}3^4$ so that $\rho\rightarrow 12$ on the sparse set of $t$-values with $Q(t)$ square (note that $Q(1)=4=2^2$, indeed the whole $ABQXY$-system was scaled by 2 to force this).
This is mentioned (in 11.5) in the above citation for Granville's rank heuristic, though for the latter there is a more recent version specified to rank 21 (rather than rank 7 in quadratic twist families).