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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?

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    $\begingroup$ Maier's theorem shows the Cramer probabilistic model can lead to false assumptions. $\endgroup$
    – Sylvain JULIEN
    Commented Aug 21, 2018 at 16:21
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    $\begingroup$ 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. $\endgroup$
    – Greg Martin
    Commented Aug 21, 2018 at 16:46
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    $\begingroup$ "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." (sciencedirect.com/science/article/pii/S0022314X08000504) $\endgroup$
    – Sam Hopkins
    Commented Aug 21, 2018 at 16:47
  • $\begingroup$ @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. $\endgroup$
    – Craig Feinstein
    Commented Aug 21, 2018 at 19:52
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    $\begingroup$ related: Are there examples of conjectures supported by heuristic arguments that have been finally disproved? $\endgroup$
    – Wolfgang
    Commented Aug 21, 2018 at 20:22

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