What about 1729 ?

So, what about this number 1729, what's so special about it? Or is it?

Many years ago I was reading the book Gödel, Escher, Bach, an eternal golden braid by Douglas Hofstadter, a book that really stretched my mind and shaped my world view.

Just one of the many dialogues in this book with the fictional characters Achilles, the Tortoise and the Crab mentions the number 1729 and hints that it's a very special number. In the following chapter there is an anecdote of the British mathematician G. H. Hardy when he visited Indian mathematician Srinivasa Ramanujan who was in a hospital . It would have went like this:

I remember once going to see him when he was ill at Putney. I had ridden in taxi cab number 1729 and remarked that the number seemed to me rather a dull one, and that I hoped it was not an unfavourable omen. "No," he replied, "it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways."

1729 = 1³ + 12³ = 9³ + 10³

It turns out 1729 is the first integer number that can be written in two different ways as the sum of two kubes. When you search this number on Wikipedia, you'll find there's a whole lot of other unique properties of this seemingly dull number.

And so I was in the middle of re-reading GEB for the umpteenth time in 1996/1997 when I discovered it was possible for an individual to register domain names on the internet. Being a bit of a nerdy computer guy, I went ahead and registered my first domain.

1729.net

Registered many other domain names since, but this was the first and therefore special. So I kept it, mostly unused. Now that I've decided to write some blogposts once in a while, this domain name felt like a perfect fit for it. And so here we are, right in the first post.

So, this was my first blog. More posts will undoubtedly follow as I intend to write no less then 1729 meaningful posts. At one post a week, that should keep me busy for a bit over 33 years. Maybe then I will run out of interesting ideas or I'll just die. If not, I'll have to continue writing 2375 more post, 45 more years, for obvious reasons.

Afterthought

Just to satisfy my curiosity and for fun I wrote some simple programs to explore properties of similar numbers to 1729, also known as the Hardy-Ramanujan number. It is also called a taxi-cab number

Here is an example of such a small script; it calculates a list of the next taxi-cab numbers, where all can be expressed as a³ + b³ = c³ + d³, with a, b, c and d smaller then 1000:

#!/usr/bin/perl



for $i (1..1000) {

    $kubes->[$i] = int $i ** 3;

    for $j (1..$i) {

        $sum = $kubes->[$i] + $kubes->[$j];

        if (defined $sums->{$sum}) {

            $sums->{$sum} = $sums->{$sum} . " or $i^3 + $j^3";

            $hrnrs->{$sum} = $sums->{$sum};

        } else {

            $sums->{$sum} = "$i^3 + $j^3";

        }

    }

}



for $i (sort { $a <=> $b } keys %$hrnrs) {

    print "$i = $hrnrs->{$i}\n";

}

Here's the first lines of output of this script:

1729 = 10^3 + 9^3 or 12^3 + 1^3

4104 = 15^3 + 9^3 or 16^3 + 2^3

13832 = 20^3 + 18^3 or 24^3 + 2^3

20683 = 24^3 + 19^3 or 27^3 + 10^3

32832 = 30^3 + 18^3 or 32^3 + 4^3

39312 = 33^3 + 15^3 or 34^3 + 2^3

40033 = 33^3 + 16^3 or 34^3 + 9^3

46683 = 30^3 + 27^3 or 36^3 + 3^3

64232 = 36^3 + 26^3 or 39^3 + 17^3

65728 = 33^3 + 31^3 or 40^3 + 12^3

110656 = 40^3 + 36^3 or 48^3 + 4^3