Quotients of Sums and Differences of Perfect Squares 1 1 



r [23] 



p2„{a2, . . ,^2) pnKa2, . . .,aii)-{-p,;\a2, . . .,a^_i) 

 has been computed, then 



Y P2n+i(ai, . . -.^i) pnjah • • .\2a{)—p^{a\ a^^ 



p2n+i(.a2, . . .,^2) pn{.a% ■ • . ,2a\)—pn\a-2, . . .,az) 



may be readily calculated, for by [i] 



pn{ai, . . .,2^1) =/„(«!, . . .,aA)p(2ai, . . .,a^) 



/'„(<a!2, . . .,2ai) =pn(a2, . . .,ajk)p(2ai, . . .,«^) 



4-i5»«(a2, . . .,ajt-i)p(2ai, . . .,a^_i) 



+pn(a2, . . .,ak-\)p{az, . . .,«^_i) J 

 The next higher quotient 



y __ A«+2(^i. . • -.^^i) __ p'n+i(ah • ■ .,ak)~'rp'^n-{-i(ai, . . .,ai-\) 

 P2n+2(^2, . . .,^2) p^„+i(a2, . . .,ak)-{-p'\+i{a2, . . .,^-4-1) 



is obtained by employing the recurrence formulae 



pn+iia^, . . .,ak) ='pn{a\, . • .,2ax)p(^a2, . . .,ak) 



-\-pniai, . . .,a2)p{a-3, . . .,ai) 



Pn+l(a2, . . .ydk) =Pn(.a2, . . .,2a{)p(a2, . . .,«>fe) 



+pn(^2, . . .,a2)p(az, . . .,ak) 



pn+i{ai, . . ..^^-i) =P?t(ai, ■ . .,2ai)p(a-2, . . -.^-fe-i) 



-\-pn(ai, . . .,a2)p(a3, . . .,ak-\) 



pn+-i{(i2, . . .,ak-\) =Pn{a.2, . . . ^2a-f)p{a2, . . .,ak-\) 



~rpn{(i2, . . .,a'2)p{ai, . . .,ak-i) 



Pni^i, • . .,«2) = Pni^i, . . .,a/i)p(a2, . . .,ak) 



+pniai, . . .,ak-\)p{a2, . . .,«,4-i) 



Pn(il2, . . .,«2) =Pn(a2, . . . ,ak)p(a2, . . .,ak) 



4-/n(«2, . . .,ak-i)p(a2, . . .,ak-i) 



365 



1^ [24] 



