The different ways of expressing whole numbers as sums of parts has long fascinated both professional and amateur mathematicians.
Consider, for example, the sequence of squares of whole numbers: 1, 4, 9, 16, 25, 36, and so forth. As the sequence progresses, the gaps between consecutive squares get longer and longer. Clearly, most integers are not squares of whole numbers.
Many integers can be written as the sum of two squares: 8 = 4 + 4; 10 = 9 + 1; 13 = 9 + 4; and so on. Other numbers can’t be expressed as the sum of just two squares. To get a sum that equals 6, the only squares available are 4 and 1, and these won’t do the job. Instead, it takes the sum of three squares: 6 = 4 + 1 + 1.
Indeed, most positive integers can be written as the sum of three squares. For instance, 11 = 9 + 1 + 1 and 12 = 4 + 4 + 4. On the other hand, 7 is an example of an integer that can’t be written as the sum of three squares. It takes four: 7 = 4 + 1 + 1 + 1.
Do you ever need more than four squares to express an integer? In 1770, French mathematician Joseph-Louis Lagrange (1736–1813) proved what previous mathematicians had suspected or assumed: Every positive integer is either a square itself or the sum of two, three, or four squares.
Earlier the same year, Edward Waring (1736–1798), a practicing physician and mathematics professor at the University of Cambridge, had 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 nine 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 Waring’s 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 conjectures about cubes and fourth powers by collecting data and looking for patterns.
The cubes of whole numbers comprise the sequence 1, 8, 27, 64, 125, . . . . The number 7 must be written as the sum of seven cubes (7 = 1 + 1 + 1 + 1+ 1 + 1 + 1); 15 requires eight cubes (15 = 8 + 1 + 1 + 1 + 1 + 1 + 1 + 1); 23 requires nine cubes (23 = 8 + 8 + 1 + 1 + 1 + 1 + 1 + 1 + 1); 31 requires only five cubes (31 = 27 + 1 + 1 + 1 + 1). Based on the evidence, it’s reasonable to suppose that no whole number is the sum of more than nine cubes.
| Integer | 1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
10 |
| Minimum no. of cubes | 1 |
2 |
3 |
4 |
5 |
6 |
7 |
1 |
2 |
3 |
| Integer | 11 |
12 |
13 |
14 |
15 |
16 |
17 |
18 |
19 |
20 |
| Minimum no. of cubes | 4 |
5 |
6 |
7 |
8 |
2 |
3 |
4 |
5 |
6 |
| Integer | 21 |
22 |
23 |
24 |
25 |
26 |
27 |
28 |
29 |
30 |
| Minimum no. of cubes | 7 |
8 |
9 |
2 |
3 |
4 |
1 |
2 |
3 |
4 |
Fourth powers, 1, 16, 81, 256, . . . , 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.
Waring’s conjectures stimulated a great deal of mathematical activity. In the early 19th century, Berlin mathematician Karl Gustav Jacob Jacobi (1804–1851) assigned the problem to his “computer,” an assistant who had compiled a list of the first 12,000 positive integers, each 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 a minimum of 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, but it includes no numbers greater than 8,042.
Such data collection doesn’t prove a conjecture, however. It serves only to suggest what may be true. Indeed, mathematicians took a long time to prove Waring’s original assertions, and they had to turn to very complicated methods to do so.
In 1909, the great German mathematician David Hilbert (1862–1943) took an important step in that direction by proving the generalization that for cubes, fourth powers, and all higher powers, there is some minimum number of terms that is sufficient to represent every whole number. However, his proof provided no guidance on what that number is for each power.
In 1912, Aubrey J. Kempner completed a 1909 effort by A. Wieferich to establish once and for all that every integer can be expressed as a sum of nine cubes. In 1940, S.S. Pillai showed that every integer can be expressed as a sum of 73 sixth powers.
The assertion that 37 fifth powers are sufficient was proved by Chen Jingrun in 1964. It wasn’t until 1986 that Ramachandran Balasubramanian, François Dress, and Jean-Marc Deshouillers proved that no more than 19 fourth powers are needed.
Mathematicians also tackled the related question of the number of terms required to express every sufficiently large integer as a sum of kth powers.
For example, even though every integer can be expressed as a sum of at most nine cubes, every integer larger than a certain value (probably 8,042) can be written as the sum of at most seven cubes. Observing the trend, mathematicians suspect that every sufficiently large integer can be expressed as the sum of no more than four cubes. The largest number now known not to be a sum of four cubes is 7,373,170,279,850.
Mathematicians have continued to work on various aspects of Waring’s problem and its variants. They have searched 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 and Noam D. Elkies 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 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 supported the conjecture that there is 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 resulting proofs.
“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,” the great mathematician Carl Friedrich Gauss (1777–1855) observed in 1849.
He continued, “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.”
Originally posted: July 13, 1996
Updated: July 17, 2004
