These blogs are intended and designed to show the way in which reasoning-led maths can generate striking, even sometimes major, results. In an age of awesomely microscopic, awesomely fast, digital electronics, authentic, traditional maths cannot compete with the latest hardware —if the contest is simply about the efficiency of manipulative process.  But we are not handle-turners. The essence of maths is not difficult manipulations.  Maths has made some of its most spectacular forward leaps   —like Euclid’s demonstration that the set of prime numbers is infinite—  by marshalling the full power of rational human insight… or if you prefer, the careful use of transparent logical reasoning.

This instalment offers an extraordinarily simple proof that any natural number can be expressed as the sum of a triangular number plus the difference between two triangular numbers. Carl Friedrich Gauss (1777-1855) first made his name as a young mathematician by proving that any natural number can be expressed as the sum of three triangular numbers. So here is a variant of Gauss’s result, viz. that we can replace one of the two pluses + in Gauss’s result with a subtract -.

Let n be a natural number and let T_o be the largest triangular number less than n. The result to be proved is that n can be expressed as the sum of T_o and T” – T’, where T” and T’ are triangular numbers.

Some triangular numbers may be represented with dots, as shown in the triangular number diagram below.  Special “dots” will be represented by s.

Examples of triangular numbers as arrays of dots:

                                                                                                                        o

                                                                                 o                              o   o

                                               o                         o   o                         o   o   o

                       o                 o   o                 o   o   o                   o   o   o   o

    o=1     o   o=3     o   o   o=6     o   o   o   o=10     o   o   o   o   o=15  and so on

Now any natural number, n, will either be a triangular number or not.

If it is already a triangular number, the proof being developed here will be unnecessary, because both T” and T’ will be zero.

The common case is when n is between triangular numbers and is x units above the nearest triangular number below n.

An example is shown in the next diagram, which looks at the case when n is 25 and x is 4:

                                  o

                             o   o

                        o   o   o

                   o   o   o   o

              o   o   o   o   o

         o   o   o   o   o   o

This is the triangular number 21.  The given number 25 is 4 units more than this.

So we add a diagonal sequence of 4 special dots s to this diagram which then represents the ordinary given natural number 25:

                                    o

                               o   o

                     s   o   o   o

               s    o   o   o   o

         s    o    o   o   o   o

    s   o    o    o   o   o   o     (T_m is (m+m²)/2. Here the s_s can be laid on the sloping side of the triangular array because x < m.)

If we add the two triangular numbers

                                                                         o

                                                                   o   o

                     s                                       o   o   o

               s    o                                o   o   o   o

         s    o    o                          o   o   o   o   o

    s   o    o    o     and      o   o   o   o   o   o       we get the array for 25 but the LH

                                 o

                           o    o

corner     o    o    o  is present twice over.  So the array for 25 can be represented by:

                                                                            o

                                                                     o   o

                     s                                        o   o   o

               s    o                                 o   o   o   o                                                              o

         s    o    o                          o   o   o   o   o                                                         o   o

    s   o    o    o     and      o   o   o   o   o   o       minus the D array    o   o   o.

This reasoning can be generalised to any given value of n.

In general terms the LHS D array with a side of x units is (x+x2)/2. This is T.

The middle D array is composed of n-x units.  This is T’.

The RHS D array is composed of (x-1 + (x-1)²)/2  or (x² –x)/2 units. This is T”.

So if we add the first two, and take away the third, we get: n -x +x/2 + x/2 or n.  It follows that any natural number, n, can be expressed as T + T’-T”. QED

CHRISTOPHER ORMELL September 12^th 2024.  Apologies for the lateness: this was caused by a technical problem.If you would like a free online copy of the P E R Narrative Maths Manifesto, send an email requesting this to per4group@gmail.com  Also comments on the reasoning in this or earlier blogs in this series can be submitted by email to the same address.  This includes any counter-argument submitted as a bid for the prize offered in blog 49.