The conjecture for which Christian Goldbach (1690-1764) is famous is that <<every even number can be represented as the sum of two prime numbers>>.  Even after sixty years of exploration using incredibly powerful computers, no one has ever found an exception to this rule.  So it has plenty of prima […]

Before starting this blog there are two preliminary Notes. NOTE 1 The author’s bare hands (elementary) proof of Fermat has now been on-line for four years —and during these 48 months no hint of a counter-argument has emerged. No challenge has been posted. It is a scandal that no-one in […]

Fermat, the last frontier The 20-page bare hands proof of Fermat’s conjecture which I posted on-line in 2020 in the first two of these blogs has now survived intact for almost four years —with no sign whatever of a demolishing counter-argument. I have since that time posted essays in the […]

Further Reflections on the exciting new proof of the Four Colour Conjecture using induction The new proof (See Blog 46) begins by posing the provisional hypothesis that all fully triangulated, fully weeded, networks of up to n nodes can be four coloured AND ALSO that this colouring can contain 3-coloured […]

Some Reflections on the new proof of the Four Colour Conjecture using induction The new proof (See Blog 46) begins by posing the provisional hypothesis that all fully triangulated, fully weeded, networks of up to n nodes can be four coloured AND ALSO that this colouring can contain 3-coloured circuits […]

Some Reflections on the Four Colour Conjecture A NEW INDUCTION ARGUMENT WITH 4-COLOURATION PLUS   The motif of this Blog is to pick up the pieces after the mistake in Blog 44, for which, further, profuse apologies. The heart of the problem is that this conjecture seems to cry out […]

Some Reflections on the Four Colour Conjecture. A MISTAKE HAS BEEN FOUND IN THE ARGUMENT OF BLOG 44. This will be reviewed in Blog 46. The problem is that the removal of Q and R from the network is possible given the 4-colouring established by the initial P, Ax, Cx arguments, […]

Some Reflections on the Four Colour Conjecture PART 3. A new proof of the Four Colour Conjecture. The four colour conjecture began as an observation about colouring maps with a palette of only four colours so that every boundary was represented by a change of colour. It soon turned into a […]

There is a common tendency for computerists to over-estimate the applicability of mathematical modelling to mysterious entities like the universe and the human brain* We know that tally-bundling must have been the natural precursor to arithmetic, and that it (tally bundling) may have been in use for many millennia before […]

Some Reflections on the Four Colour Conjecture Part 2 The four colour conjecture is a fascinating mathematical problem —because it is extremely easy to understand what it says. So it has a huge watcher-base. Also it is infuriatingly difficult to prove, which is puzzling (if it is a fact, which […]