38 ARITHMETICAL RECREATIONS [CH. II 



lishing the known results for many of the higher values of p 

 could not have been employed by Mersenne. It would be in- 

 teresting to discover how he reached his conclusions, which are 

 true if p does not exceed 88. Some recent writers conjecture 

 that his statement was the result of a guess, intelligent though 

 erroneous, as to the possible forms of p : I find it difficult- to 

 accept this opinion, but further discussion of the problem would 

 be out of place here*. 



Perfect Numbers f. The theory of perfect numbers de- 

 pends directly on that of Mersenne's Numbers. A number is 

 said to be perfect if it is equal to the sum of all its integral 

 subdivisors. Thus the subdivisors of 6 are 1, 2, and 3 ; the 

 sum of these is equal to. 6 ; hence 6 is a perfect number. 



It is probable that all perfect numbers are included in the 

 formula 2?- 1 (2* - 1), where 2* - 1 is a prime. Euclid proved 

 that any number of this form is perfect; Euler showed that 

 the formula includes all even perfect numbers; and there is 

 reason to believe — though a rigid demonstration is wanting — 

 that an odd number cannot be perfect, at any rate no odd 

 number less than 2,000,000 can be perfect. If we assume that 

 there are no odd perfect numbers, then every perfect number 

 is of the above form. It is easy to establish that every number 

 included in this formula (except when p = 2) is congruent to 

 unity to the modulus 9, that is, when divided by 9 leaves a 

 remainder 1 ; also that either the last digit is a 6 or the last 

 two digits are 28. 



Thus,ifp = 2, 3, 5, 7, 13, 17, 19, 31, 61, then by Mersenne's rule 

 the corresponding values of 2 p — 1 are prime ; they are 3, 7, 31, 

 127, 8191, 131071, 524287, 2147483647, 2305843009213693951; 



* In recent editions of this book I devoted a chapter to this problem and its 

 history, but having regard to the fact that it is now known that Mersenne's 

 statement is not true, the above notice suffices. References to memoirB on the 

 subject are given by L. E. Dickson, History of the Theory of Numbers, vol. i, 

 Washington, 1919. 



t On the theory of perfect numbers, see bibliographical references by 



H. Brocard, L'Interme'diaire des Mathimaticiens, Paris, 1895, vol. n, pp. 52 54- 



and 1905, vol. mi, p. 19. The first volume of the second edition of the French 

 translation of this book contains (pp. 280 — 294) a summary of the leading 

 investigation on Perfect Numbers, as also some remarks on Amioable Numbers. 



