SCIENCE 



[Vol. LVI, No. 1450 



my interest in this one. I do not know tlie 

 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 

 mathematicians since the times of the Greeks. 

 A number is perfect if, like 6 or 28, it is the 

 sum of all its divisors, unity included. Euclid 

 proved that the number 2"'{2'»+i — 1) is per- 

 fect if the second factor is prime; and Eiiler, 

 2,000 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 num- 

 ber 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 ?i for 

 which the answer is still in doubt. Tou may 

 perhaps be surprised that a question appar- 

 ently 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" — 1 

 is prime are 2, 3, 5, 7, 13, 17, 19, 31, 67, 127, 

 257; and an enormous amount of labor has 

 been expended on attempts to verify this asser- 

 tdon. There are no simple general tests by 

 which the primality of a number chosen at ran- 

 dom 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 some- 

 thing which later mathematicians have failed 

 to rediscover. The idea is a little fantastic, 

 but there is no doubt that, so long as the pos- 

 sibility 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 at- 

 tribute the discovery to the exceptional genius 

 of Format, the only mathematician of the age 

 whom any one could suspect of being hundreds 

 of years 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 unplumbed depths of mathemat- 

 ical profundity, was a conjecture based on 

 inadequate empirical evidence, and a some- 

 what unhappy one at that. It is . now 

 known that there are at least four num- 

 bers 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 discov- 

 ered so long ago as 1886, but could be ex- 

 plained with some plausibility, so long as it 

 stood alone, as a merely clerical error. But 

 when Mr. E. E. Powers, in 1911 amd 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 fact may be summed up as follows. 

 Mersenne makes fifty-^five assertions, for the 

 fifty-five primes from 2 to 257. Of these as- 

 sertions 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 comparatively small, or because they pos- 

 sess 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 

 dif&cult and in which 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, 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 uninterest- 

 ing in itself; that would be very far from the 

 truth. There are at least two intensely inter- 

 esting problems. The fii-st 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 



