Shapes and numbers

Abstract

Abstract The ‘numbers’ in the title of this chapter are actually integers, but the title is determined by the fact that well-known names for what we consider here are such things as triangular numbers, polygonal numbers, and polyhedral numbers. In this chapter we try to provide material that is either new or less well known, rather than provide what tends to be given in books on recreational mathematics. We start by considering the triangular numbers in some detail, highlighting their connection with the squares, and in particular dealing with the representation of positive integers as the sum of triangular numbers. In doing this we draw attention to the connections that exist between theorems on triangular numbers and square numbers. We point out that any primitive Pythagorean triplet also has its analogue amongst the triangular numbers. We then describe other shapes and the numbers associated with them. Next we consider the problem of determining which polygonal numbers are also square numbers or triangular numbers. For some polygonal numbers only a FB01nite number of them are squares, but for others there are an inFB01nite number. The Pell equation once again features in this problem. Finally, we give a brief account of the Catalan numbers.