The Interesting Number Paradox

Does there exist any number that doesn't have some particularity that makes it “interesting”? The so-called interesting numbers come from the fairly common habit of mathematics enthusiasts to find curious properties in certain numbers. Those that possess them are considered interesting, and those that don't, boring. The paradox we're dealing with today concerns precisely the existence (or not) of such numbers.

Think of any integer. Ready? Well! Suppose your brain, showing great ability to randomly select numerical values, has chosen 25. Is that number an interesting number? Mathematicians, and also fans of puzzles related to that branch of science, consider interesting those numbers that possess some quality that makes them unique, that elevates them above the infinitely large set of numbers that don't stand out for anything.

There is no universal criterion to determine whether a number is interesting or not, but in general, when someone says “the number x is interesting for such and such reason”, the rest of the interested quickly realize that, indeed, “x” has enough merit to belong to the club of interesting numbers.

All Numbers Are Interesting

The Interesting Number Paradox, precisely, slides down a slippery path whose base is the ambiguous property “being interesting”. Indeed, such a qualifier does not have an unambiguous mathematical entity, precise and objective enough to be used without hesitation as a valid criterion to divide a set of numbers. If one tried to divide a group of numbers using the property “being an even number”, one could quickly and clearly establish two groups, one made up of the even numbers and the other of the non-even (odd) numbers. The same occurs with the property “being a prime number” and many others. “Being interesting”, on the other hand, depends on each person's personal appreciation. Despite this, we will see that the paradox makes sense. A classic example of an interesting number is 1729. Although at first glance it has nothing special, it is the protagonist of an anecdote involving two brilliant mathematicians: the British Godfrey Harold Hardy and the Indian Srinivasa Aaiyangar Ramanujan.

Once, in a taxi (in English taxicab) in London, Hardy was struck by its car number, 1729. He must have been thinking about it because he entered the hospital room where Ramanujan was lying in bed and, with a dry “hello”, expressed his disappointment about this number. It was, according to him, a boring number, adding that he hoped it wasn't a bad omen. No, Hardy, said Ramanujan, it is a very interesting number. It is the smallest number expressible as the sum of two positive cubes in two different ways.

Indeed, as the incredible Indian calculated mentally, 1729 can be expressed as 1 cubed + 12 cubed, or 9 cubed + 10 cubed. This is a quite strange property, only shared by the so-called “taxicab numbers”, of which only the first 5 members of the list are known: 2, 1729, 87539319, 6963472309248 and 48988659276962496. For the sixth, it has only been calculated that it is less than or equal to 24153319581254312065344.

Returning to our randomly chosen number, 25, we could say that it is special because it is the smallest square (5 squared) that can be written as the sum of two squares: 3 squared + 4 squared. Suppose we don't have the ability to find a property for every natural number, and we decide to separate them into two groups, one made up of the “boring” numbers and the other of the “interesting” ones.

Imagine that the first number to which we cannot find any particularity is 33. That would automatically make it a very interesting number, since it has become “the smallest number that has no particularity”. That characteristic turns it into “interesting”, and therefore it must be moved to the other set. Once 33 is removed, surely another number has taken its place, becoming the new “the smallest number that has no particularity”, so we should also move it to the other set.

This can be repeated infinitely, and end up with a set of “interesting numbers” made up of all those we had at the beginning, and another that is empty. This forces us to conclude that there are no numbers that are not interesting. On the other hand, it is valid to ask what can be interesting about the members of a set that brings together all existing numbers. Indeed, a characteristic shared by absolutely all of them has nothing special. The paradox, despite being based on an ambiguous property, exists.

Some mathematics enthusiasts really enjoy finding what is interesting about each number. Erich Friedman, a mathematics professor at Stetson University, has compiled a list with the particularities of each of the integers between 0 and 9999.

Do you dare to find a truly boring number?