The Four Color Theorem was solved by Haken and Appel in 1976, with a proof that involved the use of computers. The current state of the argument along these lines can be found in work of Robertson, Sanders, Seymour, and Thomas. Many have found the Haken/Appel proof unsatisfying, largely because the use of computers makes it uninformative to people. While this aspect is considerably lessened in recent work, interest in a proof along different lines persists.
Howard Levi, at the end of a long and distinguished career in mathematics, spent the last years of his life working on a proof of the Four Color Theorem along algebraic lines. He died in 2002 with his proof unfinished. What he did establish was an interesting equivalent algebraic formulation of the problem, involving finite fields. Others have also produced algebraic reformulations of the problem, see the references below, but there are significant novelties in Howard's work to make it of independent interest.
Howard left notes and partially written papers that were somewhat chaotic. Paul Meyer and myself were friends of Howard of long standing, and we took it on ourselves to bring some order to this chaos. In this we were extensively assisted by Alan Hoffman, also an old friend of Howard, and especially by Don Coppersmith. We have produced the paper found here We acknowledge its imperfections---this is not our area. But we make it available as a tribute to our friend, relying on the search engines of the world to bring it to the attention of specialists, in the hope that it may be found of some use. Click here for the paper.
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