You might think this would have been because it was a problem I have been thinking about for years, that I have attempted to solve myself, that I have lost sleep on. Or perhaps because I am aware of the ramifications for mathematics, the new vistas that will be opened up, the new opportunities that will arise.

Well, I had heard of the abc conjecture. Perhaps I once even knew what it was, but I had forgotten. I have no knowledge of the subject area, the applications (if any) or the methods used.

Yet I am excited by the news of the possible proof! How is it that a development about a problem which I know nothing about, which I couldn't even describe in the most general terms, can matter to me?

Well, here (from the excellent Wikipedia article on the conjecture) is a statement of the conjecture: it asserts that the answer to the following question is "yes": "For every ε > 0, are there only finitely many triples of coprime positive integers

*a*+

*b*=

*c*such that

*c*>

*d*

^{ (1+ε)}, where

*d*denotes the product of the distinct prime factors of

*abc*?" What does this mean? Basically if the conjecture is true, then (as I understand it) it means that only exceptionally is the product of the prime factors of

*abc*significantly less than

*c*.

Now this is, frankly, quite an obscure statement about numbers, and I cannot envisage any life-changing applications. What I find wonderful is that human minds like mine can prove this statement. I cannot even begin to imagine how one would set about proving such a conjecture. I still have an instinctive belief (which, rationally, I know is rather naive) that mathematical facts are true regardless of the nature of the human brain, the laws of nature, and so on: thirteen would still be a prime if the human race had never existed, if the laws of physics were totally different, if no sentient creature had ever come into being. That a mind like mine can establish such necessary, deep facts is amazing, a glimpse of something much more true than anything else in our existence.

## No comments:

## Post a comment