20 SECTIONAL ADDRESSES. 



immediate neighbourhood of lOOOCOO, 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 enthely considerably before this 

 point. It is conceivable that the five-cube numbers also disappear, but 

 this, if it be so, is in depths where computation is helpless. 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 tliree. 



I need hardly 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, and there are infinitely many which cannot be 

 expressed by fewer. 



3. I will next raise the question whether the number 2"^— 1 is 

 prime. I said that I woijld include one question which did 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 mathematicians since the times of the Greeks. A 

 number is pei-fect if, like 6 or 28, it is the sum of all its divisors, unity 

 included. Euclid proved that the number 



is perfect if the second factor is primei; and Euler, 2,(J()0 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 onlj* 

 be so 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 wliich 2" - 1 is prime are 



2, 3, f), 7, 13, 17, 19, 31, 67, 127, 257; 



and an enornic/us amount of labour has been exjiended on attempts to 

 verify this assertion. There are no' simple general tests by which 

 the primality of a number chosen at i-aiKlo'Ui can be determined, and 

 the amount of computation required in any pai-ticular case may be 

 quite appalling. It has, however, been imagined that Mersenne 

 perhaps knew something which later mathematicians 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. 

 Eouse Ball in 1891. Mersenne, he observes, was a good mathemati- 

 cian, but not an Euler or a Gauss, and he inclines to attribute the 

 discovery to the exceptional genius of Fermat, the only mathematician 



