Mathematics: Beauty and the Beast Walter Tholen

Mathematics: Beauty and the Beast

• Walter Tholen

• York University

• Toronto

N = p1 · p2 · …. · pn + 1

• N has a prime factor p.

• That prime factor p cannot be one of p1, p2, …, pn,

• for if it were, p would not only be a divisor of N,

• but also of N – 1 = p1 · p2 · …. · pn : impossible!

Twin primes

• 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, …

• Are there infinitely many twin primes?

• J. G. van der Corput (1939):

• There are infinitely many triples of primes in arithmetic progression.

• Ben Green and Terence Tao (2004): There exist sequences of primes in

• arithmetic progression of any given length.

Luc Lemaire:

• Luc Lemaire:

• How should we remember Carl Wilhelm Ferdinand, Duke of Brunswick? In my dictionary, he is described as a duke soldier who was beaten by the French in Valmy, then again in Jena. But I must say I looked only in a French dictionary. Still, an uninspiring notice. But one day, he got a report from a school teacher that a young boy seemed remarkably gifted in mathematics. The boy was the son of a poor gardener and bricklayer, so his future should have been rather bleak. But the Duke liked mathematics, saw the boy and was convinced by his obvious talent (if not by his good manners). Thus he supported his studies and career throughout his life. The boy’s name was Carl Friederich Gauss, and we owe to him (and the Duke) the Gauss law of prime numbers, the Gauss distribution in probability, the Gauss laws of electromagnetism, most of non-Euclidean geometry, and the Gauss approximation in optics.

• Obviously, we need more Dukes of Brunswick in our governments!

Saunders Mac Lane (1909 – 2005)

• The progress of mathematics is like the difficult exploration of possible

• trails up a massive infinitely high mountain, shrouded in a heavy mist

• which will occasionally lift a little to afford new and charming

• perspectives. This or that route is explored a bit more, and we hope

• that some will lead higher up, while indeed many routes may join and

• reinforce each other.