CH - n ] ARITHMETICAL RECREATIONS 37 



and so on. Similarly 41 = 5 a +4 2 , 41 2 = 40 2 + 9 2 , 41 s = 236 2 +115 2 , 

 and so on. Propositions such as the one just quoted may be 

 found in text-books on the theory of numbers and therefore 

 lie outside the limits of this work, but there are one or two 

 questions in higher arithmetic involving points not yet 

 quite cleared up which may find a place here. I content 

 myself with the facts and shall not give the mathematical 

 analysis. 



Primes. The first of these is concerned with the possibility 

 of determining readily whether a given number is prime or not. 

 No test applicable to all numbers is known, though of course 

 we can get tests for numbers of certain forms. It is difficult to 

 believe that a problem which has completely baffled all modern 

 mathematicians could have been solved in the seventeenth 

 century, but it is interesting to note that in 1643, in answer 

 to a question in a letter whether the number 100895,598169 

 was a prime, Fermat replied at once that it was the product of 

 898423 and 112303, both of which were primes. How many 

 mathematicians to-day could answer such a question with 

 ease? 



Mersenne's Numbers. A curious assertion (in this case 

 only partially correct) about the prime or composite character 

 of numbers of the form 2^ — 1 (which we may denote by N) is 

 to be found in Mersenne's Gogitata Physico-Mathematica, pub- 

 lished in 1 644. In the preface to that work a statement is made 

 about perfect numbers, which implies that the only values of 

 p not greater than 257 which make iV prime are 1, 2, 3, 5, 7, 13, 

 17, 19, 31, 67, 127, and 257. Some years ago I gave reasons for 

 thinking that 67 was a misprint for 61. Until 1911, no error 

 in this corrected statement was established, and it was gradually 

 verified for all except sixteen values of p. In 1911, however, it 

 was proved, that N~ was prime when p — 89, and three years 

 later that it was prime when p — 107 : two facts at variance with 

 Mersenne's statement. The prime or composite character of iV 

 now remains unknown for only ten values of p, namely 139, 149, 

 157, 167, 193, 199, 227, 229, 241, and 257. 



We may safely say that the methods used to-day in estab- 



