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Are there any calculations or results that have similar answers and when compared numerically look the same, but in actual fact after so much precision, the answers diverge from each other?

An example I could think of would be a fractional approximation for $\pi$ such as $\frac{245850922}{78256779}$. If one was unaware of $\pi$'s irrationality then they might think (through numerical "proof") that they were the same thing. Obviously this is a trivial and silly example so I am looking for examples that are more major and may even have been of a surprise to some.

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  • $\begingroup$ You could search for "lies my calculator told me"! $\endgroup$ – Maesumi Dec 1 '13 at 22:31
  • $\begingroup$ This seems to fit in with this question from the "most votes" list. There are some answers there that directly fit this question. This answer in particular is excellent. $\endgroup$ – Eric Thoma Dec 1 '13 at 22:32
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    $\begingroup$ Related and very interesting mathoverflow.net/questions/131528/… $\endgroup$ – clark Dec 1 '13 at 22:32
  • $\begingroup$ Thanks to Maesumi. $\endgroup$ – Git Gud Dec 1 '13 at 22:36
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Ramanujan gave the identity

$$e^{\pi \sqrt{163}} = 262537412640768743.99999999999925007...$$

Martin Gardner once claimed as a hoax that $e^{\pi \sqrt{163}}$ is an integer, which apparently confused a lot of people: when they tried to disprove the claim on their calculators, they found that $e^{\pi \sqrt{163}}$ was indeed an integer - but only because their calculators were not precise enough!

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  • $\begingroup$ Strictly speaking, the value of $e^{\pi\sqrt{58}}$ is found in Ramanujan's Notebooks, but not $e^{\pi\sqrt{163}}$. However, in one section, he uses $d=19,43,67,163$ so he may have been aware of this near-integer property as well. Strange though that he didn't write it down. $\endgroup$ – Tito Piezas III Oct 28 '17 at 11:48
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You might enjoy the answers to this question over on MathOverflow that catalogs some examples of the phenomenon of "eventual counterxamples," particularly the Borwein integrals.

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