3 82 



NA TURE 



[September 16, 1922 



geometry that fits the Eai ts the best, for anything that 

 mathematicians have to say. There may be three 

 dimensions in this room and five next door. As a 

 professional mathematician, I have no idea ; I can 

 onl) a k some competent physicist to instruct me in 

 the facts. 



The function of a mathematician, then, is simply 

 to observe the facts about his own intricate system 

 of reality, that astonishingly beautiful complex of 

 logical relations which forms the subject-matter of 

 his si ience, as if he were an explorer looking at a 

 distant range of mountains, and to record the results 

 of his observations in a series of maps, each of which is 

 a branch of pure mathematics. Many of these maps 

 have been completed, while in others, and these, natur- 

 ally, are the most interesting, there are vast uncharted 

 regions. Some, it seems, have some relevance to the 

 structure of the physical world, while others have no 

 such tangible application. Among them there is 

 perhaps none quite so fascinating, with quite the same 

 astonishing contrasts of sharp outline and mysterious 

 shade, as that which constitutes the theory of numbers. 



The number system of arithmetic is, as we know too 

 well, not without its applications to the sensible world. 

 The currency systems of Europe, for example, conform 

 to it approximately ; west of the Vistula, two and 

 two make something approaching four. The practical 

 applications of arithmetic, however, are tedious beyond 

 words. One must probe a little deeper into the subject 

 if one wishes to interest the ordinary man, whose 

 taste in such matters is astonishingly correct, and 

 who turns with joy from the routine of common life 

 to anything strange and odd, like the fourth dimension, 

 or imaginary time, or the theory of the representation 

 of integers by sums of squares or cubes. 



ft is impossible for me to give you, in the time at 

 my command, any general account of the problems 

 of the theory of numbers, or of the progress that has 

 been made towards their solution even during the 

 last twenty years. I must adopt a much simpler 

 method. I will merely state to you, with a few words 

 of comment, three or four isolated questions, selected 

 in a haphazard way. They are seemingly simple 

 questions, and it is not necessary to be anything of a 

 mathematician to understand them ; and 1 have 

 chosen them for no better reason than that I happen 

 to be interested in them myself. There is no one of 

 them to which I know the answer, nor, so far as I 

 know, does any mathematician in the world ; and 

 there is no one of them, with one exception which I 

 have included deliberately, the answer to which any 

 one of us would not make almost any sacrifice to know. 



i. When is a number the sum of two cubes, and what 

 is the number of its representations ? This is my first 

 question, and first of all I will elucidate it by some 

 examples. The numbers 2 = i 3 +i 3 and q = 2 3 +i 3 are 

 sums of two cubes, while 3 and 4 are not : it is ex- 

 ceptional for a number to be of this particular form. 

 The number of cubes up to 1,000,000 is 100, and the 

 number of numbers, up to this limit and of the form 

 required, cannot exceed 10,000, one-hundredth of the 

 whole. The density of the distribution of such numbers 

 tends to zero as the numbers tend to infinity. Is there, 

 I am asking, any simple criterion by which such 

 numbers can be distinguished ? 



NO. 2759, VOL. I IO] 



Again, 2 and 9 are sums of two cubes, and can be 

 expressed in this form in one way only. There are 

 numbers so expressible in a variety of different ways. 

 The least such number is 1729. which is i2 3 +i 3 and 

 also io 3 + 9 3 . It is more difficult to find a number 

 with three representations ; the least such number is 



1 75,959,000 = 56o 3 + 7o 3 = 552 3 + i98 3 = 525 3 + 3i5 3 . 

 One number at any rate is known v/ith four representa- 

 tions, namely, 



19X363510 3 



(a number of 18 digits), but I am not prepared to 

 assert that it is the least. No number has been 

 calculated, so far as I know, with more than four, but 

 theorv, running ahead of computation, shows that 

 numbers exist with five representations, or six, or any 

 number. 



A distinguished physicist has argued that the 

 possible number of isotopes of an element is probably 

 limited because, among the ninety or so elements at 

 present under observation, there is none which has 

 more isotopes than six. I dare not criticise a physicist 

 in his own field ; but the figures I have quoted may 

 suggesf to you that an arithmetical generalisation, 

 based on a corresponding volume of evidence, would 

 be more than a little rash. 



There are similar questions, of course, for squares, 

 but the answers to these were found long ago by Euler 

 and by Gauss, and belong to the classical mathematics. 

 Suppose, for simplicity of statement, that the number 

 in question is prime. Then, if it is of the form 4;/; + 1, 

 it is a sum of squares, and in one way only, while if it 

 is of the form 4111 + 3 it is not so expressible ; and 

 this simple rule may readily be generalised so as to 

 apply to numbers of any form. But there is no 

 similar solution for our actual problem, nor, I need 

 scarcely say, for the analogous problems for fourth, 

 fifth, or higher powers. The smallest number known 

 to be expressible in two ways by two biquadrates is 



635318657 = is8 4 +59 4 = i34 4 + 133 4 ; 



and I do not believe that any number is known ex- 

 pressible in three. Nor, to my knowledge, has the 

 bare existence of such a number yet been proved. 

 When we come to fifth powers, nothing is known at 

 all. The field for future research is unlimited and 

 practically untrodden. 



2. I pass to another question, again about cubes, but 

 of a somewhat different kind. Is every large number 

 (every number, that is to say, from a definite point 

 onwards) the sum of five cubes ? This is another 

 exceptionally difficult problem. It is known that every 

 number, without exception, is the sum of nine cubes ; 

 two numbers, 23 (which is 2.2 3 + 7.i 3 ) and 239, actually 

 require so many. It seems that there are just fifteen 

 numbers, the largest being 454, which need eight, and 

 121 numbers, the largest being 8042, which need seven ; 

 and the evidence suggests forcibly that the six-cube 

 numbers also ultimately disappear. In a lecture 

 which I delivered on this subject at Oxford I stated, 

 on the authority of Dr. Ruckle, that there were two 

 numbers, in the immediate neighbourhood of 1,000,000, 

 which could not be resolved into fewer cubes than 

 six ; but Dr. A. E. Western has refuted this assertion 

 by resolving each of them into five, and is of opinion, 



