|Ivars Peterson's MathTrek|
July 15, 1996
Clearly, most numbers are not squares of whole numbers. Some, however, can be written as the sum of two squares: 13 = 9 + 4. Others cannot. To get a sum of 6, the only squares available are 4 and 1, and that's not enough. But three squares are sufficient: 6 = 4 + 1 + 1. Similarly, 7 can't be written as the sum of three squares, but it can be written as the sum of four squares: 7 = 4 + 1 + 1 + 1. Two squares suffice for 8 (4 + 4); 9 is itself a square; 10 = 9 + 1; 11 = 9 + 1 + 1; 12 = 9 + 1 + 1 + 1 or 4 + 4 + 4.
You might suspect that, at some point, four squares would no longer be enough to express a given whole number. This reasonable supposition was overturned in seventeenth century, when Pierre de Fermat proved that every positive whole number can be expresed as a sum of at most four squares.
In 1777, Edward Waring, a practicing physician and mathematics professor at the University of Cambridge, conjectured that something similar could be proved for cubes, fourth powers, and so on. He stated, without proof, that it would take the sum of at most 9 cubes or 19 fourth powers to express any whole number.
Waring had a reputation as a brilliant mathematician deeply concerned about fundamental concepts in mathematics. His interest was not in practical applications but in illuminating the nature of mathematics itself. Unfortunately, the clumsiness and impenetrability of his writings kept him from achieving recognition for much of his pioneering work. His name, to the extent that it is known at all today, is attached to problems concerning the sums of powers of whole numbers.
Waring likely arrived at his conjecture by experiment. The cubes of whole numbers consist of the sequence 1, 8, 27, 64, É. The number just before 8 must be written as the sum of seven cubes: 7 = 1 + 1 + 1 + 1+ 1 + 1 + 1 (7 cubes); 15 requires 8 cubes; 23 requires 9 cubes; 31 requires only 5 cubes (31 = 27 + 1 + 1 + 1 + 1). Based on the pattern for squares, it's not unreasonable to suppose that no whole number is the sum of more than nine cubes.
Waring's idea stimulated a great deal of mathematical activity, and key parts of his conjecture remain unproved.
In the early nineteenth century, Berlin mathematician Carl Gustav Jacob Jacobi assigned the problem to his "computer," an assistant who compiled a list of the first 12,000 positive integers, expressed as the sum of the smallest possible number of cubes. In that list, the only number other than 23 that requires nine cubes is 239. Fifteen numbers require eight cubes: 15, 22, 50, 114, 167, 175, 186, 212, 213, 238, 303, 364, 420, 428, and 454. The list of numbers requiring seven cubes is much longer, and it includes no numbers greater than 8,042.
Such data collection, however, doesn't prove the conjecture. It serves only to suggest what may be true. It was not until this century that mathematicians proved, using very complicated methods, that every number beyond a certain point can be written as the sum of just seven cubes.
Fourth powers, 1, 16, 81, 256, É, show similar behavior: 15 can be written as the sum of 15 fourth powers, 31 requires 16 fourth powers, 47 requires 17, 63 requires 18, and 79 requires 19.
In 1909, the great German mathematician David Hilbert approached the problem from a different angle. He proved the generalization that for cubes, fourth powers, and all higher powers, there is some minimum number of terms that will suffice to represent every whole number. However, he did not determine what that number is for each power. So, Waring's problem was solved in principle but not in practice.
In 1925, British mathematicians John E. Littlewood and G.H. Hardy took another step toward a full answer by proving that all numbers beyond a certain point are the sums of at most 19 fourth powers. To settle the case for fourth powers, mathematicians merely needed to check every whole number up to that point. But this point is such a huge number that even today's electronic computers can't fill in the gap.
Since then, many mathematicians have studied the problem and its variants, substantially lowering the bounds for powers between four and ten. They have also looked for other patterns involving powers. They have studied sums of mixed powers (whole numbers as the sum of two squares and a cube, for instance) and sums of powers in which both positive and negative integers are allowed.
In 1995, Irving Kaplansky of the Mathematical Sciences Research Institute in Berkeley, Calif., and Noam D. Elkies of Harvard University proved that any integer can be expressed as the sum of two squares and a cube, when positive and negative integers are allowed. Using a computer, Kaplansky and William C. Jagy of the University of California, Berkeley then showed that the analogous situation for a square and two cubes holds in the range from -4,000,000 to 2,000,000. Additional computations support the conjecture that there are a finite number of exceptions to the rule that all whole numbers can be expressed as the sum of a square and two cubes, using only positive integers.
In their search for patterns, Kaplansky, Jagy, and others have explored a wide range of possible combinations, and there's much left to investigate and ponder -- fertile ground for the dedicated amateur.
This work continues the tradition of mathematical experiment to help discover patterns, suggest conjectures, and develop new theorems. What's striking in the case of the arithmetic of whole numbers is the gulf between the apparent simplicity of the raw materials and the immense complexity and delicacy of the proofs required.
In 1849, the great mathematician Carl Friedrich Gauss observed: "The higher arithmetic presents us with an inexhaustible storehouse of interesting truths -- of truths, too, which are not isolated, but stand in the closest relation to one another, and between which, with each successive advance of the science, we continually discover new and wholly unexpected points of contact.
"A great part of the theories of arithmetic derive an additional charm from the peculiarity that we easily arrive by induction at important propositions, which have the stamp of simplicity upon them, but the demonstration of which lies so deep as not to be discovered until after many fruitless efforts; and even then it is obtained by some tedious and artificial process, while the simpler methods of proof long remain hidden from us."
Copyright © 1996 by Ivars Peterson.
Jagy, William C., and Irving Kaplansky. 1995. Sums of Squares, Cubes, and Higher Powers. Experimental Mathematics 4(No. 3): 169-173.
Mahoney, Michael Sean. 1994. The Mathematical Career of Pierre de Fermat, 1601-1665 (2nd ed.). Princeton, N.J.: Princeton University Press.
Rademacher, Hans, and Otto Toeplitz (Herbert Zuckerman, trans.). 1990. The Enjoyment of Mathematics: Selections from Mathematics for the Amateur. New York: Dover (originally published by the Princeton University Press, 1957).
Schroeder, M.R. 1984. Number Theory in Science and Communication: With Applications in Cryptography, Physics, Biology, Digital Information, and Computing. New York: Springer-Verlag.
Stewart, Ian. 1986. The Waring experience. Nature 323(Oct. 23): 674.
Biographical information about Edward Waring is available at http://mitlns.mit.edu/~bruen/waring.html..\