Never a Dull Number
THOSE FASCINATING NUMBERS. Jean-Marie De Koninck. xviii + 426 pp. American Mathematical Society, 2009. $49.
It all begins with a taxi ride. In 1918 the Cambridge mathematician G. H. Hardy goes to visit his protégé Srinivasa Ramanujan and mentions that the number of his taxicab was rather dull: 1,729. Not dull at all, Ramanujan replies; 1,729 is the smallest number that can be written as the sum of two cubes in two different ways (123 + 13 and 103 + 93).
From this famous anecdote springs a mathematical joke. If 1,729 is not a dull number, then which numbers are dull? In particular, what is the smallest number that has no interesting traits—nothing to make it stand out in the endlessly receding line of undistinguished integers? The joke is this: Whatever number you choose as the smallest dull number immediately becomes highly interesting because it’s the smallest dull number.
Those Fascinating Numbers, a collection of numerical lore by Jean-Marie De Koninck, can’t escape the pseudo-paradox of dull numbers. The book is essentially a list of the counting numbers, presented in the usual order, with notes on curious facts about each of them: 1 is the only number that divides all the others; 2 is the only even prime number; 3 is the smallest triangular number.
Clearly, this can’t go on forever. And so one is led to ask: What is the first number that De Koninck skips over in his enumeration—the smallest number for which he could find no facts worth mentioning? I’ll give the answer at the end of this review; for now I’ll just note that the first omission in the sequence follows an integer identified as “the seventh number whose sum of divisors is a perfect square.” You’re welcome to work that out, if you’d like.
What makes a number interesting? This is a question each of us must answer for ourselves; tastes differ. De Koninck, a mathematician at the University of Laval in Quebec, gives much attention to the divisors of numbers and to patterns in their decimal digits. For example, the entry for 153 points out that it is one of just five numbers equal to the sum of the cubes of its digits; and 26 is distinguished as the smallest number that is not a palindrome but whose square is a palindrome. These are not properties that excite my personal sense of wonder. On the other hand, I was intrigued by the observation that 70 is “the only number whose square corresponds to the number of cannonballs one can pile up in a pyramidal shape (702 = 12 + 22 + . . . + 242).” And I’m glad to know that 1,368 is the number of ways you can fold a 3-by-3 sheet of stamps.
This is a book for browsing rather than sustained reading. It would make an excellent online resource, since searching and following cross-references would be more convenient. And it’s the kind of book that ought to be kept on hand in barrooms or coffee shops where the numerati hang out, an authority for settling arguments and wagers over the second-smallest number that can be written as the sum of two cubes in two different ways (4,104) or the smallest dull number according to De Koninck (95).
Brian Hayes is Senior Writer for American Scientist. He is the author most recently of Group Theory in the Bedroom, and Other Mathematical Diversions (Hill and Wang, 2008).