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Several open conjectures are widely believed to be true due to strong heuristical evidence. Examples include Goldbach's conjecture , Collatz's conjecture and the Riemann hypothesis.

Are there examples of conjectures that turned out to be false despite of a strong heuristical evdidence to be true ?

I know that there are conjectures with very large smallest counterexamples , but I wonder whether there were additional reasons (except that a counterexample wasn't found for a long time) to believe the conjecture.

Klangen
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Peter
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    Interesting question, +1 –  Mar 13 '18 at 21:52
  • There was never really an hypothesis posited, but Godel's theorem might be a good example. – Steven Alexis Gregory Mar 13 '18 at 22:11
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    As I heard it, the set of prime numbers was strongly believed not to be a diophantine set, up until it was proved that actually all recursive sets are diophantine. With the reasoning being - how could there possibly be a polynomial in $n$ variables (with integer coefficients) whose positive values are exactly the prime numbers?! – Daniel Schepler Mar 13 '18 at 22:15
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    Cramer's model incorrectly predicts that primes are uniformly distributed at polylogarithmic scales, but this is now known to be wrong: https://en.wikipedia.org/wiki/Maier%27s_theorem – Wojowu Mar 14 '18 at 14:51
  • @Wojowu Really bad news because this model is widely used to estimate the number of primes, for example , in a range. Is the fact it is false critical in practical applications ? – Peter Mar 14 '18 at 14:57
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    I am not working in the field, but I wouldn't think this is too critical - this just says that primes are not too regularly distributed in short ranges. I think the model is still expected to be essentially correct for longer ranges (say, at lengths at least $x^c,c>0$). – Wojowu Mar 14 '18 at 15:01
  • @Peter , Number theorist here, Wojowu is essentially correct. The break down of Cramer's model is while I would hesitate to call it well understood something we understand. In general, on short scales primes don't behave as much like a pseudorandom sequence as on a large scale. The intuition behind this is that if two numbers are near to each other than knowing that one is prime restricts the prime factors of another a decent amount whereas if they are far away then it is much harder for the information in some sense to transmit the distance. – JoshuaZ Sep 26 '19 at 22:41
  • I'm not sure to what degree this was formalized as a conjecture, so I'm writing this as a comment, but available numerical evidence once suggested that $\pi(x) > \operatorname{li}(x)$, where $\pi$ is the prime counting function and $\operatorname{li}(x) := \int_0^x \frac{dt}{\log t}$ is the logarithmic integral function. Littlewood showed in 1914 that in fact the inequality fails for infinitely many $x$. – Travis Willse Sep 27 '19 at 01:47
  • It's not so surprising that numerical evidence indicated otherwise: The inequality holds for all numbers $< 10^{19}$, and the best known upper bound for the first point of failure is large, about $1.4 \cdot 10^{316}$. – Travis Willse Sep 27 '19 at 01:47

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There is a conjecture that the number of primes between $x$ and $x+y$, inclusive, is never more than the number of primes between $2$ and $2+y$, inclusive. It seems reasonable, since anyone can see that the primes thin out as you go up, and it has never been disproved, but forty years ago it was proved to be in conflict with another unproved conjecture that has even better heuristic evidence, so it's now believed to be false.

Gerry Myerson
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They were rarely called conjectures, but when General Relativity was proposed by Einstein it was expected that solutions would be "physically reasonable". Much later Godel discovered an exact solution, which allowed time travel into the past among other things. Later it was thought that singularities were a consequence of the high degree of symmetry of solutions, such as the Schwarzschild black hole solution, but later Hawking and others proved that singularities arise under fairly general conditions. "Cosmic Censorship", the idea that singularities are always hidden behind event horizons and never visible to us (although the big bang is essentially a visible singularity) is still a conjecture of classical General Relativity. Hawking showed that when quantum effects are included, black holes emit radiation (contrary to the belief that nothing can escape) and a black hole can "evaporate", but may create a visible singularity as it dissapears.

tippy2tina
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