September 16, 1922] 



NA TURE 



I believe, that the six-cube numbers have disappeared 

 entirely considerably before this point. It is con- 

 ceivable that the five-cube numbers also disappear, 

 but this, if it be so, is probably in depths where 

 computation is helpless. The four-cube numbers must 

 certainly persist for ever, for it is impossible that a 

 number 9/7 + 4 or 977 + 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 com- 

 plete. About the squares there is no mystery ; every 

 number is the sum of four squares, and there are infin- 

 itely many numbers which cannot be expressed by fewer. 



3. I will next raise the question whether the number 

 2 1 " - 1 is prime. I 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 my interest in this one. I do not 

 know the answer, and I do not care greatly what it is. 



The problem belongs to the theory of the so-called 

 " perfect " numbers, which has exercised mathe- 

 maticians since the times of the Greeks. A number 

 is perfect if. like 6 or 28. it is the sum of all its divisors, 

 unitv included. Euclid proved that the number 



2OT(2 m + l_ I ) 



is perfect if the second factor is prime ; and Euler, 

 2000 years later, that all even perfect numbers are of 

 Euclid's form. It is still unknown whether a perfect 

 number can be odd. 



It would obviously be most interesting to know 

 generally in what circumstances a number 2" - 1 is 

 prime. It is plain that this can be so only if n itself 

 is prime, as otherwise the number has obvious factors ; 

 and the 137 of my question happens to be the least 

 value of n for which the answer is still in doubt. You 

 may perhaps be surprised that a question apparently 

 so fascinating should fail to arouse me more. 



It was asserted by Mersenne in 1644 that the only 

 values of n, up to 257, for which 2 n - 1 is prime are 



2, 3. 5, 7, !3. !7.- i9>3!, 6 7, 12 h 2 57 ; 

 and an enormous amount of labour has been expended 

 on attempts to verify this assertion. There are no 

 simple general tests by which the primality of a number 

 chosen at random can be determined, and the amount 

 of computation required in any particular case may be 

 appalling. It has, however, been imagined that 

 Mersenne perhaps knew something which later mathe- 

 maticians have failed to rediscover. The idea is a 

 little fantastic, but there is no doubt that, so long as 

 the possibility remained, arithmeticians were justified 

 in their determination to ascertain the facts at all 

 costs. " The riddle as to how Mersenne's numbers 

 were discovered remains unsolved," wrote Mr. Rouse 

 Ball in 1891. Mersenne, he observes, was a good 

 mathematician, but not an Euler or a Gauss, and he 

 inclines to attribute the discovery to the exceptional 

 genius of Fermat, the only mathematician of the age 

 whom any one could suspect of being hundreds of 

 3'ears ahead of his time. 



These speculations appear extremely fanciful now, 

 for the bubble has at last been pricked. It seems now 

 that Mersenne's assertion, so far from hiding un- 

 plumbed depths of mathematical profundity, was a 



NO. 2759, VOL. I 10] 



conjecture based on inadequate empirical evidence, 

 and a somewhat unhappy one at that. It is now 

 known that there are at least four numbers about 

 which Mersenne is definitely wrong ; he should have 

 included at any rate 61, 89, and 107, and he should 

 have left out 67. The mistake as regards 61 and 67 

 was discovered so long ago as 1886, but could be 

 explained with some plausibility, so long as it stood 

 alone, as a merely clerical error. But when Mr. 

 R. E. Powers, in 1911 and 1914, proved that Mersenne 

 was also wrong about 89 and 107, this line of defence 

 collapsed, and it ceased to be possible to take Mersenne's 

 assertion seriously. 



The facts may be summed up as follows. Mersenne 

 makes fifty-five assertions, for the fifty-five primes 

 from 2 to 257. Of these assertions forty are true, 

 four false, and eleven still doubtful. Not a bad result, 

 you may think ; but there is more to be said. Of the 

 forty correct assertions many, half at least, are trivial, 

 either because the numbers in question are com- 

 paratively small, or because they possess quite small 

 and easily detected divisors. The test cases are those 

 in which the numbers are prime, or Mersenne asserts 

 that they are so ; there are only four of these cases 

 which are difficult and in which the truth is known ; 

 and in these Mersenne is wrong in ever)' case but one. 



It seems to me, then, that we must regard Mersenne's 

 assertion as exploded ; and for my part it interests 

 me no longer. If he is wrong about 89 and 107, I 

 do not care greatly whether he is wrong about 137 

 as well, and I should regard the computations necessary 

 to decide as very largely wasted. There are so many 

 much more profitable calculations which a computer 

 could undertake. 



I hope that you will not infer that I regard the 

 problem of perfect numbers as uninteresting in itself ; 

 that would be very far from the truth. There are at 

 least two intensely interesting problems. The first 

 is the old problem, which so many mathematicians 

 have failed to solve, whether a perfect number can be 

 odd. The second is whether the number of perfect 

 numbers is infinite or not. If we assume that all 

 perfect numbers are even, we can state this problem 

 in a still more arresting form. Are there infinitely 

 many primes of the form i n -\1 I find it difficult to 

 imagine a problem more fascinating or more intricate 

 than that. It is plain, though, that this is a question 

 which computation can never decide, and it is very 

 unlikely that it can ever give us any data of serious 

 value. And the problem itself really belongs to a 

 different chapter of the theory, to which I should like 

 next to direct your attention. 



4. Are there infinitely many primes of the form 

 n 2 +i? Let me first remind you of some well-known 

 facts in regard to the distribution of primes. 



There are infinitely many primes ; their density 

 decreases as the numbers increase, and tends to zero 

 when the numbers tend to infinity. More accurately, 

 the number of primes less than x is, to a first ap- 

 proximation, 



x 



log*' 



The chance that a large number n, selected at 



random, should be prime is, we may say, about ,— — . 



