In 1770 Lagrange proved the beautiful theorem that every positive integer is the sum of 4 integer squares and on July 10th 1796 Gauss made the first entry in his celebrated notebook.
Num = Δ + Δ + Δ
which records the fact that he had proved that every positive integer is the sum of 3 triangular numbers.
Since that time a great deal of work has been done on the representation of positive integers by the sum of k-th powers (Waring's Problem) and also representation by the sum of polyhedral numbers, and so on. These are natural generalisations of the Lagrange and Gauss theorems in the upwards direction. Some time ago it occurred to me that there might be a result in the downward direction also, that is, it might be possible to find a sequence of positive integers generated by a reasonably natural and non-trivial function such that any positive integer could be expressed as the sum of 2 elements of the sequence. The trivial solutions are those in which the sequence has positive density (i.e. contains a non-zero fraction of all the integers), such as (1,2,3,4,5,10,15,20,...).
We are therefore looking for a sequence generated by a function f(n) where f(n) is of higher order than n but of lower order than n2. The most natural function satisfying these conditions seemed to be f(n) = [ns] where 1 < s < 2 and [x], as usual, denotes the integer part of x.
I asked Ann Donaldson (Mrs A H J Walter), of the Atlas Laboratory, to write a program to generate [ns] + [ms] for various rational values of s in the interval [1,2) and we found, almost at once, that all the integers up to 10,000 can be represented by [n4/3] + [m4/3] whereas the sequence [n3/2] + [m3/2], for example, fails to represent 23 integers below 10,000, the first being 17. We also studied the slightly different sequence [ns + ms] and the results were essentially the same. The numerical evidence strongly supported the Conjecture: Every positive integer can be represented in the form [n4/3] + [m4/3] or in the form [n4/3 + m4/3].
It also seemed possible that every sufficiently large positive integer can be expressed in the form [n3/2] + [m3/2] since the largest of the 23 numbers which failed was 1577, so that all the integers from 1578 to 10,000 certainly can be represented.
The evidence for the conjecture was very good but for about a year I made no attempt to prove it, because I had no idea how to do so. The classical problems were all concerned with representation as sums of integers generated by functions such as f(x) = x2 or f(x) = ½x(x+1). Functions such as these not only have important arithmetical properties but they can also be introduced as exponents in infinite series such as
(which is essentially a theta function)
-a move which opens up a whole armoury of weapons. Unfortunately a function such as f(x) = [x4/3] plays no central role in analysis or number theory, and so such powerful weapons are denied to us.
Shortly after I moved to Cardiff I took a fresh look at the problem and realised that if we could obtain a reasonably accurate idea of the size of the fractional part of ns (usually denoted by {ns}) it might be possible to prove the conjecture. Of course on average {ns} = ½ but what I needed was an estimate of the error term, viz: if
then, how large is R(N)? This looked more feasible as I recalled that a number of theorems on the sums of fractional parts of functions were known, and that the best source for such a result was the works of I. M. Vinogradov.
It turned out that the Vinogradov had proved a theorem relating to
for a certain class of functions f(x). As things stood I couldn't apply the theorem directly because the particular function of xs which I had to use didn't satisfy Vinogradov's conditions. However it was possible to apply Vinogradov's theorem to a different function g(x), related to xs over a limited range (N1- ε,N) and by arranging ε to be a function of N and letting N tend to infinity (and ε to zero simultaneously) I was able to prove the following:
Theorem. Every sufficiently large integer, N, can be expressed in the form N = [ns] + [ms], n and m being positive integers and s being any number in the interval (1, 4/3). (A full proof is being published elsewhere).
The great surprise was the entirely natural appearance of the exponent s = 4/3. The theorem is proved for an expression of the type N = [ns] + [ms] where s satisfies a certain inequality and it happens that as N tends to infinity so s tends to 4/3.
Because of the nature of the proof there is no hope of replacing the phrase every sufficiently large integer in the theorem by every integer.
In order to improve the exponent from 4/3 to, say, 3/2 a really major improvement in Vinogradov's basic theorem would be required and there are very good reasons for believing that this would require some entirely new approach.
The total machine time required for this work was only a few minutes and Ann wrote the program as a training exercise, but without the evidence from Atlas I would never have attempted to prove such a theorem. Looking back over the past 7 years I find that Atlas has played an essential role in five of the papers which I have written: in a sense it has been a co-author, and I am profoundly grateful.
