Volume 32


In this issue, we bring you four articles which illustrate the diversity of Mathematics. In the article on Fermat’s Last Theorem, Professor Michael Cowling has shown how a long-standing problem about integers has recently been solved by looking at the graphs of functions which appear to be quite irrelevant.

David Rowe, a Year 12 student at Barker College, rang us recently with the following question. We thought that it would be of interest to our readers.

Most of us will have a tree in our homes over Christmas. To the mathematician, a tree is a special type of graph.

Our aim is to find a simple formula for the area of a polygon whose vertices are
$$A_1(x_1,y_1),\quad A_2(x_2,y_2),\quad \cdots\quad A_n(x_n,y_n),$$
joined in that order.

It was recently found that some versions of the Intel Pentium processor possessed a flaw in the floating point divide unit. This generated considerable interest in the algorithm used in this chip to perform the division of two real numbers.

Some 3000 years ago, the ancient Egyptians knew that the triangle with sides $3$, $4$ and $5$ is a right-angled triangle. And of course, they also knew the related fact that $9+16=25$, i.e., that $3^2+4^2=5^2$.

Q.985 For what values of the positive integer $n$ is

  1. $5n + 2$
  2. $7n + 2$

a perfect square?

Q.975 For which real numbers $x$ is it true that
$$ [5x] = [3x] + 2[x] + 1 ? $$
Here $[x]$ denotes the greatest integer less than or equal to $x$; for example, $[\pi] = 3$.