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There are examples of conjectures in which one can use probabilistic heuristic reasoning to show that they are very likely to be true. For instance, Freeman Dyson used probabilistic heuristic reasoning to show that "it never happens that the reverse of a power of two is a power of five." But he also believes that this statement is impossible to prove - "because there is no deep mathematical reason why it has to be true".

But there is at least one example where probabilistic heuristic reasoning fails, namely one of two Hardy-Littlewood's Conjectures.

1) The k-tuple conjecture, which states that the asymptotic number of prime constellations can be computed explicitly.

2) $\pi(x+y) \leq \pi(x) + \pi(y)$, where $\pi$ is the prime counting function.

Probabilistic heuristic reasoning can be used to argue that both of these conjectures are true, yet in 1974, Ian Richards proved that these two conjectures are incompatible with each other.

Are there any other examples in which probabilistic heuristic reasoning fails?

Craig Feinstein
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    Maier's theorem shows the Cramer probabilistic model can lead to false assumptions. – Sylvain JULIEN Aug 21 '18 at 16:21
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    In (2), I would argue that the original probabilistic heuristic was flawed; the corrected heuristic (proposed after it was shown that (1) and (2) are inconsistent) predicts that $\pi(x+y) \le \pi(x) + 2\pi(\frac y2)$, which I think is believed to be true. – Greg Martin Aug 21 '18 at 16:46
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    "Cohen–Lenstra heuristic and roots of unity" by Gunter Malle: "We report on computational results indicating that the well-known Cohen–Lenstra–Martinet heuristic for class groups of number fields may fail in many situations. In particular, the underlying assumption that the frequency of groups is governed essentially by the reciprocal of the order of their automorphism groups, does not seem to be valid in those cases. The phenomenon is related to the presence of roots of unity in the base field or in intermediate fields." (https://www.sciencedirect.com/science/article/pii/S0022314X08000504) – Sam Hopkins Aug 21 '18 at 16:47
  • @GregMartin, I don't see how the original probabilistic heuristic is flawed, even if in retrospect it is. After all, the density of primes decreases as the numbers get larger. Even the possibility that it is flawed is shocking. – Craig Feinstein Aug 21 '18 at 19:52
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    related: Are there examples of conjectures supported by heuristic arguments that have been finally disproved? – Wolfgang Aug 21 '18 at 20:22
  • When examined carefully, the heuristic actually says that density of primes increases as the absolute value of numbers get larger. (Here negative numbers are allowed to be prime.) Therefore the interval of length $y$ with the largest number of primes should be the interval whose elements are smallest in absolute value — that is, the interval $(-\frac y2,\frac y2]$. – Greg Martin Aug 21 '18 at 22:38
  • Actually, negative numbers are not allowed to be prime in that particular conjecture. See the link. – Craig Feinstein Aug 22 '18 at 01:01
  • @Wolfgang, I was not aware of that link. It seems that my question is only about probabilistic arguments, while that question includes all types of heuristics. I am not sure what to do. Should I delete this question? – Craig Feinstein Aug 22 '18 at 01:04
  • @GregMartin Do you have a reference? I have not heard of that refined heuristic. – Wojowu Aug 22 '18 at 06:50
  • @CraigFeinstein Funny coincidence (serendipity??), as I've asked that question less than a week ago!! I was convinced that you'd seen it and wanted to narrow it down to your specific field of interest. Anyway, no need to delete it, and that would kill the specific discussion going on in here. – Wolfgang Aug 22 '18 at 07:58
  • @Wojowu, I am now reading the paper by Ian Richards that I linked to above, and it appears that it addresses the issue that Greg Martin raised. Anyway, the paper is a good read. – Craig Feinstein Aug 22 '18 at 16:17
  • Tao describes a probabilistic argument that the orbits of the Collatz conjecture should decrease in magnitude, and thus ultimately pass through $1$. It is of course conceivable that the Collatz conjecture is false (and possibly the probabilistic argument correct), but all simulation studies suggest that the conjecture is true. Admittedly, this is no proof, but it suggests that there may be a hidden flaw in the statistical argument. (See: https://terrytao.wordpress.com/tag/collatz-conjecture/ ) – David G. Stork Aug 23 '18 at 21:04
  • That argument has been in the literature for quite some time. It is a good heuristic probabilistic argument. If ever someone proves the Collatz Conjecture false, then that would be an answer to my question. – Craig Feinstein Aug 24 '18 at 00:21

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