Quotients of Sums and Differences of Perfect Squares 

 since by [2] 



and 



Likewise 

 p(a2, . . 



.,ak--[) — p(ai, . . .,aji') = p(ai, . . ..ajt-z). 



,^2) = -7- 1 p^ia2, . . .,ak) — ^^(«i, . • .,ak-2) 



ak 



V 



In the quotient of /(^i, . . .,a\) by /(«2, 

 out, and we have 



,,^2), ak divides 



II. Every number whose order is odd can be expressed as a 

 quotient of differences of squares of positive integers. 



If the continuant /(ai, . . .,ai) is of odd order, each of the other 

 continuants [19] is of odd order. The middle element is ak in the 

 first, third, fifth and 2a, in the second, fourth, etc. The steps that 

 led to Theorem II., lead likewise to each of the equations: — 



— , L':>M, order odd, 



L _ p(ai, . . .,ai) _pKa h ■ ■ .,ak)—p '^ i^h - • -^^^-2) 

 M p{^a-i, . . .,^2) />^(«2, . . .,ak)—p^(a2, . . ■,«/fe-2) 



pi(ah . • •,<^i) /^(ai, . . .,2aj)—p^(ai, . . .,as) 



pi(a2, . . .,«2) />^l«2, . . .,2ai) — p^{^a% . . ,,az) . 



pi{a\, . . .,ai) p^{a\, . . .,ak)—p i- (ai, . . .,ak-2 ) 



~~ p-i{,a% . . .,a-2) P^{a% . . .,ak)—Pi^(a2, . . .,^^-2) 



p2n(ah . . .,a\) pn{a\, • • .,ak)—p,t^(a\, ■ • .,ak-2) 



pi7i{a-i, . . .,a-2) ~ pn^{a2, . . .,ak)-—pn{_a2, . . .,ak-o) 



pin+\(au---,ai) Pnjai, . . .,2ai)—Pn^(ai, . . .,^3) 



Pzn+i{a2,...,a2') pniai, . . .,2ai)—Pn\a2, . . .,as) 



> [21] 



.-^63 



