Kummer, Ernst Eduard (1810-1893)

German mathematician who introduced ‘ideal numbers’ in the attempt to prove Fermat's last theorem. His research into systems of rays led to the discovery of the fourth-order surface known as the Kummer surface.

Kummer was born in Sorau (now Zary, Poland) and studied at Halle. He was professor at Breslau 1842-55, when he was appointed to professorships at the University of Berlin and the Berlin War College.

Ideal numbers is one of the most creative and influential ideas in the history of mathematics. With their aid Kummer was able to prove, in 1850, that the equation:

xl + yl + zl = 0

was impossible in non-zero integers for all regular prime numbers (a special type of prime number related to Bernoulli numbers). He was then able to determine that the only primes less than 100 that are not regular are 37, 59, and 67. For many years Kummer continued to work on the problem and he was eventually able to prove that the equation is impossible for all primes l < 100.

The Kummer surface can be described as the quartic which is the singular surface of the quadratic line complex and involves the very sophisticated and complicated concept of this surface as the wave surface in space of four dimensions.

Kummer also made an important contribution to function theory by his investigations into hypergeometric series.