Every number can be written as the sum of nine cubes.
Every number can be written as the sum of nineteen fourth powers.
Waring’s claims were correct, and cannot be improved because there are numbers that actually require nine cubes and nineteen fourth powers, such as
for cubes:
for fourth powers:
Waring asked whether these ideas can be extended to higher powers:
Waring’s problem: For each positive integer k, does there exist a number g(k) such that every number can be written as the sum of g(k) kth powers?
For example, , by Lagrange’s four-square theorem, and the results just asserted tell us that and . Other known values are for sums of fifth powers, and for sums of sixth powers. In 1909, the German mathematician David Hilbert answered Waring’s question in the affirmative by proving that there is such a number g(k) for each value of k.
What can we say about g(k)? Around 1772, Johann Albrecht Euler, Leonhard’s eldest son, suggested that
where ⌊x⌋ is the integer part of x: for example, .
Note that,
so this bound gives the correct value for g(k) in these two cases. It has since been proved to give the right answer for all values of k up to 471.6 million, and with all this evidence it’s believed to be correct in every case, although this has never been proved.
But we can say more. When , it turns out that only two numbers (23 and 239) actually need nine cubes, and that only a finite number need eight cubes. So we can say that,
From some point onwards, every number can be written as the sum of just seven cubes.
It is also believed, although this has never been proved, that the number of cubes can be reduced further, possibly even to four.
For fourth powers it can be proved that,
From some point onwards, every number can be written as the sum of just sixteen fourth powers.
Corresponding results can be proved for higher powers.Fermat’s last theorem
We conclude this chapter with a brief account of one of the most famous achievements in number theory—the proof of Fermat’s last theorem.
In our discussion of Pythagorean triples, we saw how to find integer solutions of the Diophantine equation
Can we likewise find integer solutions to the equations
and in general,
We can make a couple of general observations.
First, there are always ‘trivial’ solutions, in which one of the numbers a, b, and c is 0: for example,
In what follows we’re interested only in positive solutions.
Secondly, suppose that we knew that there were no integer solutions of the equation . Then we could deduce that the same is true when the exponent is any multiple of 3. For example, the equation could then have no integer solutions, because we can rewrite it as
which we’ve assumed to have no solutions. So, when investigating solutions of the equation , we can restrict our attention to the cases when n is 4 or a prime number.
Fermat became interested in the problem while reading a Latin translation of Diophantus’s Arithmetica that had been published by Claude Bachet de la Méziriac in 1621 (see Figure 26). In the section on Pythagorean triples Fermat added a now-famous marginal comment:
On the other hand, it is impossible for a cube to be written as a sum of two cubes or a fourth power to be written as a sum of two fourth powers or, in general, for any number which is a power greater than the second to be written as a sum of two like powers. I have a truly marvellous proof of this proposition which this margin is too narrow to contain.
26. Bachet’s translation of Diophantus’s Arithmetica.
This is often called ‘Fermat’s last theorem’, because it became the last of Fermat’s assertions to be proved—but it should perhaps have been called ‘Fermat’s conjecture’:
Fermat’s last theorem. The equation has no positive integer solutions when .
Most mathematicians believe that Fermat couldn’t have had a valid proof of his conjecture. Many attempts have since been made to find one, and the belief is that, if he really had found a correct argument, then it would surely have been rediscovered over the ensuing years.
Fermat himself proved that the equation can have no integer solutions. To do so, he invented a method of proof known as the method of infinite descent, showing by an algebraic argument that if this equation (or, more precisely, one closely related to it) had a solution in positive integers, then there would be another solution that involved smaller numbers than before. By repeating this argument over and over again, he’d then get smaller and smaller positive solutions—but this cannot happen indefinitely. This contradiction shows that the original solution couldn’t have existed in the first place.
More than a century later, in 1770, Euler produced a proof for the case , but it had a gap in it that was subsequently filled by Legendre.
Major advances were made in the 1820s by Sophie Germain, a self-taught mathematician who made several substantial contributions to number theory and to the theory of elasticity. Interested in primes p for which is also prime (such as 2, 3, 5, 11, and 23), she proved several results concerning them—for example, that if , then one of a, b, and c must be divisible by , and one of them must be divisible by p2. Then Legendre and Lejeune Dirichlet proved the theorem when , using her results, and in 1835 Gabriel Lamé of Paris proved it for
