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Along with Richard Greenblatt, he may be considered to have founded the hacker community, and holds pride of place in the Lisp community.
Affiliated with the MIT AI Lab during his prime, he performed many programming feats, mainly in, but not restricted to, computational mathematics.
Gosper made major contributions to the Macsyma computer algebra system at MIT, later working with Symbolics and Maxima, Inc. on the greatly improved commercial versions. Also while at MIT, he did a lot of work on the MacLisp system.
He became intensely interested in the Game of Life shortly after John Horton Conway had proposed it, conjectured on the existence of infinitely growing patterns, and offered a reward for an example. Gosper was the first to find such a pattern, and won the prize. He is also the originator of the hashlife algorithm that can speed up the computation of Life patterns by many orders of magnitude.
In the 1970s, Gosper moved to California, where he lectured at Stanford and helped Donald Knuth write volume II of The Art of Computer Programming.
He played the leading role in compiling the HAKMEM.
He is also noted for his work on continued fraction representations of real numbers, and for suggesting the algorithm (which bears his name) for finding closed form hypergeometric identities.