Kenneth I. Appel, who helped usher the venerable mathematical proof into the computer age, solving a longstanding problem concerning colors on a map with the help of an I.B.M. computer making billions of decisions, died on April 19 in Dover, N.H. He was 80.

The cause was esophageal cancer, which was diagnosed in October, his son Andrew said.

Since the time of Euclid and Pythagoras, proofs of mathematical theorems had consisted of long strings of equations or geometric notations that any mathematician could read and quibble with, all marching logically, step by step, toward a conclusion. But the proof that Dr. Appel and a colleague, Wolfgang Haken, established in 1976 was of a different order.

Their conclusion, that four colors would suffice for any map, depended on 1,200 hours of computer time — the equivalent of 50 days — and 10 billion logical decisions all made automatically and out of sight by the innards of an I.B.M. computer at the University of Illinois in Urbana.

Hailed in some circles, including this newspaper, as “a major intellectual feat,” the proof shepherded computers toward a greater role in higher math. But it made many mathematicians uneasy; they worried about computer bugs and wondered how they could check or understand a “proof” they could not see. And it ignited a long-running debate about what constitutes a mathematical proof.