# When Have Numerology and Computational Experimentation Been Successful?

When has numerology been successfully used in math and science? The Monstrous Moonshine conjecture led to a Fields medal for Borcherds. Balmer's formula for hydrogen spectra led to the Bohr model of the atom.

We could extend this to general computational experimentation. For example, the Birch-Swinnerton-Dyer conjecture was originally formulated based on sketchy computational results. Gauss guessed the law of quadratic reciprocity and the prime number theorem from his calculations too. Are there other interesting or instructive examples?

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I would say that most conjectures in number theory are supported by computational experimentation. What do you mean by "numerology"? –  Pete L. Clark Jun 18 '10 at 18:40
I think "numerology" here means conjectures that are supported by less computational evidence than someone with ridiculously high standards would like. –  Michael Lugo Jun 18 '10 at 18:43
Mirror symmetry? –  Steve Huntsman Jun 18 '10 at 18:56
–  Steve Huntsman Jun 18 '10 at 19:30
Community wiki? –  Grétar Amazeen Jun 18 '10 at 19:34

Three examples from the theory of dynamical systems (in a broad sense).

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Searching for "experimental mathematics" will give you entire books and a journal's worth of examples. This seems to be too broad a question for MathOverflow. However, I will give one example: The Riemann hypothesis. In Borwein and Bailey's book they give evidence that Riemann arrived at this conjecture by means of calculating the first few zeros.

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Kepler is a very interesting case, because his work relied almost entirely on experimentation (of good, scientific kind) $\textit{and}$ he sometimes invoked astrological and numerological ideas. For example, in spite of correctly concluding that it is sectorial area, not arc length, that describes the motion of planets, he went on to "prove" that the circumference of the ellipse is $\pi(a+b).$ –  Victor Protsak Jun 18 '10 at 23:21