What is called experimental mathematics is the exploration of mathematics and the possibe truth of theorems not by general proof, but by checking special cases using computer programs.
A general proof of Riemann hypothesis remains famously open, but computer experiment so far confirms it, see there.
A general proof of the volume conjectures remains open (though various special cases do have proof), but computer experiment (such as using SnapPy) so far confirms (and motivates) these conjectures.
The image of the canonical homomorphism $\beta$ from the Burnside ring of a finite group to the representation ring over some field is partly known only by computer experiment, see there.
David H. Bailey, Jonathan M. Borwein, Experimental Mathematics: Examples, Methods and Implications, Notices of the AMS, Volume 52, Number 5, 502-514. (pdf)
David H. Bailey, Jonathan M. Borwein, Neil J. Calkin, Roland Girgensohn, D. Russell Luke, Victor H. Moll, Experimental Mathematics in Action. (pdf)
Herbert S. Wilf, Mathematics: An Experimental Science. (pdf)
See also
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