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?
 A: 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.
A: There is this MO thread Experimental Mathematics about experimental mathematics, with more than 40 answers.
A: Perhaps it would be useful to record some of the failures of numerology as well, e.g., Kepler's attempt to model the solar system by inserting the five Platonic solids among the six planets that were known in his day. Kepler's eventual success with ellipses could be viewed as numerology as well, as the theoretical basis for it had to wait for Newton. 
A: Three examples from the theory of dynamical systems (in a broad sense).


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*Edward Lorenz' almost accidental discovery of  the first example of a strange attractor and the butterfly effect.


      
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*The Fermi-Pasta-Ulam numerical experiments  which for the first time exhibited the soliton-like dynamics in a nonlinear system.  

*Mitchell Feigenbaum's experiments with iterations of the logistic map which led to the discovery of universality in nonlinear systems.
