October 13, 1922] 



SCIENCE 



403 



and of the form required, can not exceed 

 10,000, one hundredth of the whole. The den- 

 sity of the distribution of such numbers tends 

 to ze-ro as the numbers tend to infinity. Is 

 there, I am asking, any simple criterion by 

 which such numbers can be distinguished? 



Again, 2 and 9 are sums of two cubes, aad 

 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 12^ + 1^ and also 10' + 9^. It 

 is more difficult to find a number with three 

 representations; the least such number is 

 175,959,000 = 560' + 70= = 552= + 198' = 

 525' + 315'. One number at any rate is 

 known with four representations, namely, 

 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, run- 

 ning ahead of computation, shoAvs 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 suggest to 

 you that an arithmetical generalization, based 

 on a corresponding volume of evidence, would 

 be more than a little rash. 



There are simikr 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 nmnber in 

 question is prime. Then, if it is of the form 

 4m -f 1, it is a smn of squares, and in one 

 way only, while if it is of the form 4ot + 3 it is 

 not so expressible; and this simple rule may 

 readily be generalized so as to apply to num- 

 bers of any form. But there is no similar solu- 

 tion for our actual problem, nor, I need scarce- 

 ly 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 = 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. When we come to 

 fifth powers, nothing is known at all. The 

 field for future research is unlimited and prac- 

 tically 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 dif- 

 ficult problem. It is known that every number, 

 without exception, is the sum of nine cubes; 

 two iBumbers, 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 lec- 

 ture which I delivered on this subject at Ox- 

 ford I stated, on the authority of Dr. Ruckle 

 that there were two numbers, in the immediate 

 neighborhood 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, I believe, that the six-cube numbers 

 have disappeared entirely considerably before 

 this point. It is conceivable that the five-cube 

 numbers also disappear, but this, if it be so, is 

 probably in depths where computatiion is help- 

 less. The four-cube numbers must certainly 

 persist for ever, for it is impossible that a 

 number 9n + 4 or 9n + 5 should be the sum 

 of three. 



I need scarcely add that there is a similar 

 problem for every higher power. For fourth 

 powers the critical number is 16. There is no 

 case, except the simple case of squares, in 

 which the solution is in any sense complete. 

 About the squares there is no mystery; every 

 number is the sum of four squares, and there 

 are infinitely many numbers which can not be 

 expressed by fewer. 



3. I will next raise the question whether the 

 number 2'-^' — 1 is prime. 1 said that I would 

 include one question which does not interest 

 me , particularly ; and I should like to explain 

 to you the kind of reasons which damp down 



