Theorem of the Month

www.shef.ac.uk/~puremath/theorems/all.html

The University of Sheffield Pure Mathematics Department has put its best foot forward in this attempt to demonstrate the inherent interest of mathematics through some of its most intriguing theorems such as the Fundamental Theorem of Algebra, Godel’s Theorem, the Four-colour Theorem and several others.

One theorem is featured, the remainder can be accessed through a list on the main page. Although this site suffers from inconsistencies in style, numerous typographical problems, and occasional bad exposition, it has enough saving features to make it worth a visit by students of mathematics, as well as those with an interest in some of the more accessible theorems.

What You See

It is, after all, a university site, so we weren’t expecting fireworks. Beside a prim blue bar that provides links to other pages of the university’s site, there is a list of theorems or topics under the heading Other Theorems of the Month: Pythagorean Triples and the Congruent Number Problem; Lagrange’s Four-square Theorem; Bezout’s theorem; Solution of Cubics and Quartics; The Date of Easter; The Fundamental Theorem of Algebra; Godel’s Theorem; The Four-colour Theorem; The Arithmetic-geometric Mean Inequality, Cantor’s Diagonal Argument; Fermat’s Last Theorem; The Tactical Voting Problem; Is (indecipherable) an Integer?; The Prime Number Theorem; Pythagoras Animated (the Applet bombed, then froze our computer).

We will not describe all of these topics, just a few for flavour. Pythagorean Triples and the Congruent Number Problem promises a formula for Pythagorean triples, proves a theorem that provides the formula, but then never explains to the reader how to use the formula. Lagrange’s Four-square Theorem states that any positive integer n can be written as the sum of four squares. The proof is built up gradually, with a stop at n being the sum of two squares. Bezout’s theorem is simple enough when you state it: Two polynomial curves of degree m and n respectively intersect in nm points. After a catalogue of exceptions , wherein the “points are re-interpreted as points at infinity, double points, etc, the ground shifts beneath our feet to complex projective space, where the theorem is “proved.” Solution of Cubics and Quartics explains that every polynomial of degree 1, 2, 3, and 4 can be solved by radicals (roots of coefficients) but that some polynomials of degree 5 cannot be so solved. The Fundamental Theorem of Algebra that every non-constant polynomial with complex coefficients has a complex root is never actually proved. But there’s an insightful stop at cubic polynomials where existence of the root is proved using the Intermediate Value Theorem of Analysis. Three alternate proofs are sketched. Of course, Godel’s Theorem is never proved but there’s a nice demonstration of a non-computable function (marred by typographical accidents) and an important link to a translation of Godel’s original paper. We’ll stop at the Four -colour Theorem, where a plethora of attractive and interesting diagrams makes us wonder if we’ve entered a new site. There is much historical information here and an explanation of why the problem is harder than it first appears to be.