A.— MATHEMATICS AND PHYSICS, 19 



l.lie whole. The density of the distribution of suuh numbers tends (o 

 zero as tlie number tends to infinity. Is there, I am asking, any simple 

 criterion by which such immbers can be distinguished? 



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, wfiich is 12^ + 1^ 

 and also 10^ + 9^ It is more difficult to find a number with three 

 representations ; the least such number is 



175959000 = 560' + 70'' = 552' + 198" = 525' + 315'''. 

 One number at any rate is known with four representations, viz. 



19 X 363510' 

 (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 theory, 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 obsei^atioo, there is none which has more 

 isotopes than six. I dare not criticise a physicist in liis own field; 

 but the figures I have quoted may suggest 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 v\^ere 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-hi + I, it 

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

 4m. + 3 it is not so expressible; and this simple rule may readily be 

 generahsed so as tO' apply to numbers ol any form. But there is no 

 similar solution ior our actual problem, nor, I need hardly say, for the 

 analogous problems for fourth, fifth, or higher powers. The smallest 

 number known toi be expressible in two' ways by two biquadrates is 



635318657 = 158' + 59' = 134' + 133' ; 

 and I do not believe that any number is known expressible in three. 

 Nor, to my knowledge, has the bare existence of such a number yet 

 been proved. ^Yhen 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 mimber (every number, that is to say, 

 from a definite point onwards) Uie sum of five cubes ' This is another 

 exceptionally difficult problem. It is known that every number, with- 

 out exception, is the sum of nine cubes ; two numbers, 23 (which is 

 2.2^ + 7.1^) 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. Euckle, that there were two numbers, in the 



