Quotients of Sums and Differences of Perfect Squares 7 



Every non-quadratic rational number can be expressed as a 

 quotient of tzvo simple reciprocal^ continuants, whose elements 

 arc positive integers, and of zvhich one is formed by omitting the 

 initial and final elements of the other. 



Moreover, since every quadratic surd can be expressed as a 

 continued fraction, with unit numerators and positive integral de- 

 nominators, in one way only, the representation of non-quadratic 

 rational numbers by means of quotients of continuants is unique, 

 there is a one-to-one correspondence between the doma;n of non- 

 quadratic rational numbers and the totality of simple reciprocal 

 continuants with a single cycle of positive integral elements. 

 To each number N of this domain corresponds one definite con- 

 tinuant which we may call its continuant, and each number of 

 the domain belongs to one or the other of two classes according 

 as its continuant is of even or odd order. We may attach the 

 order of its continuant to the number itself, and speak of num- 

 bers as having odd or even order, which of course has nothing 

 to do with the odd- or evenness of the numbers themselves ; 

 6, 19, 28, 51 are of an odd order, 10, 13, 58, 97 of even order. 



If we admit continuants of more than a single cycle of ele- 

 ments, the one-to-one correspondence ceases to exist. For any 

 pure quadratic surd is represented equally well by each of the 

 forms 



L 



■i 



iM * * 



= (ai,a2, . . .,2ai, . . .,a2,2ai), 



* * 



== {ai,a2, . . .,2ai, . . .,2ai, . . .,a2,2ai), 



* * 



= (^1,^2, . . .,2ai, . . .,2ai, . . .,2ai, . . .,«2,2(2i), 



* * 



etc., 

 consequently to the same number L/M corresponds each of the 

 continuants 



p{a\,a% . . .,ai,a2), 

 P(ai,a2, . ■ ■,2ai, . . .,^2,^1), 

 P(ai,a2, . . .,2ai, . . .,2a\, . . .,a2,ai), I [19] 



piax.CLi 2a\, . . .,2ai, . . .,2ai, . . .,a2,ai), | 



etc., J 



1 A reciprocal continuant is one in which ak = dn-k for every value of k, 



361 



