CH. Il] ARITHMETICAL RECREATIONS 43 



is sufficient to confine ourselves to cases where n is a prime. 

 A prize* of 100,000 marks has been offered for a general proof, 

 to be given before 2007. 



Naturally there has been much speculation as to how Fer- 

 mat arrived at the result. The modern treatment of higher 

 arithmetic is founded on the special notation and processes 

 introduced by Gauss, who pointed out that the theory of 

 discrete magnitude is essentially different from that of con- 

 tinuous magnitude, but until the end of the last century the 

 theory of numbers was treated as a branch of algebra, and such 

 proofs by Fermat as are extant involve nothing more than 

 elementary geometry and algebra, and indeed some of his 

 arguments do not involve any symbols. This has led some 

 writers to think that Fermat used none but elementary 

 algebraic methods. This may be so, but the following remark, 

 which I believe is not generally known, rather points to the 

 opposite conclusion. He had proposed, as a problem to the 

 English mathematicians, to show that there was only one 

 integral solution of the equation a? + 2 = y 3 : the solution 

 evidently being x = 5, y = 3. On this he has a notef to the 

 effect that there was no difficulty in finding a solution in 

 rational fractions, but that he had discovered an entirely new 

 method — sane pulcherrima et subtilissima — which enabled him 

 to solve such questions in integers. It was his intention to 

 write a work J on his researches in the theory of numbers, but 

 it was never completed, and we know but little of his methods 

 of analysis. I venture however to add my private suspicion 

 that continued fractions played a not unimportant part in his 

 researches, and as strengthening this conjecture I may note 

 that some of his more recondite results — such as the theorem 

 that a prime of the form 4ra + 1 is expressible as the sum of 

 two squares — may be established with comparative ease by 

 properties of such fractions. 



* UlntermSdiaire des MatMmaticiens, vol. xv, pp. 217—218, for references 

 and details. 



t Format's Diophantus, bk. vi, prop. 19, p. 320; or Brassinne'a Pricu, 



p. 122. 



% Format's Diophantus, bk. iv, prop. 31, p. 181 ; or Brassinne's Prielt, p. 82. 



