The Euclid – Euler Theorem

William Dunham
Truman Koehler Professor of Mathematics
Muhlenberg College

As the last number theoretic proposition in his Elements, Euclid proved that if 2n 1 is prime, then 2n–1(2n 1) is perfect (i.e., equal to the sum of its proper divisors). Twenty centuries (!) later, Euler established the partial converse: that any even perfect number must have this structure. Using original sources, we shall see how Euclid and Euler joined forces in exploring this rare but intriguing type of number.

BIOGRAPHY

William Dunham, who received his B.S. (1969) from the University of Pittsburgh and his M.S. (1970) and Ph.D.(1974) from Ohio State, is the Truman Koehler Professor of Mathematics at Muhlenberg College.

Originally trained in general topology, Dunham became interested in the history of mathematics. He has directed NEH-funded summer seminars on math history at Ohio State and has spoken on historical topics at national and regional mathematics meetings as well as at the Smithsonian Institution, on NPR’s “Talk of the Nation: Science Friday,” and on the BBC. His expository writing has been recognized with the MAA’s George Polya Award in 1991 and the MAA’s Trevor Evans Award in 1997, and he received the Award for Distinguished College or University Teaching from the MAA’s Eastern Pennsylvania and Delaware (EPADEL) section in 1994.

Dunham combined his historical and mathematical interests by authoring Journey Through Genius: The Great Theorems of Mathematics (John Wiley, 1990). A second book, The Mathematical Universe (Wiley, 1994), received the American Association of Publishers’ award as the Best Mathematics Book of 1994. His most recent work, Euler: The Master of Us All, was published by the Mathematical Association of America in 1999.

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