A.— MATHEMATICSAND PHYSICS. 21 



of the age whom anyone could suspect of being huiich-cds of years ahead 

 of his time. 



These speculations appeal- extremely fanciful, for tlie bubble has 

 at last been i:)ricked. it seems now that Mei'seune's assertion, so 

 far from hiding unplumbed depths of.malliematical profundity, was a 

 conjecture based on inadequate empincal evidence, and a rather 

 unhappy onsi at tha.t. 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 1U7, and he should have left out 67. 

 The mistake as regards 61 and 67 was discovered as long ago as 1886, 

 but could be explained with some plausil)ility, so long as it stood alone, 

 &s a merely clerical error. But when Mr. E. 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 smnmed 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 tiivial, either because the numbers in 

 (]uestion are compara.tively small, or because they possess quite small 

 and easily detected divisors. The test cases are those in whicli 

 IMersenne asserts the numbers in question tO' lie prime ; there are only 

 four of these cases which are difficult and in w hich the truth is known ; 

 and in these Mersenne is wrong in every 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 or not, and I sho'uld 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 caii be odd. The second is whether the 

 number of perfect numbers is infinite or not. If we assume that all 

 perfect numbers are infinite, w^e can state this prololem in a still more 

 an-esting form. Are there mflnitclij many pr'niiea of the form 2"- 1? 

 I find it hard to imagine a problem more fascinating or more terribly 

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

 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- + l? 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. 



