lO 



Robert E. Morits 



This result may be stated thus ; 



III. Every number whose order is odd may he expressed in 

 an unlimited number of different ways as a quotient of differ- 

 ences of squares of positive integers. 



If the continuant /)(«!, . . . ,a{) is of even order, the succession oi 

 continuants [19] have alternately even and odd order. The 

 middle elements of the first, third, fifth, etc., is a/., while the 

 second, fourth, etc., have each 2ai for the middle element. Hence 

 we have successively 



L 



■jr^, V^M, order even, 



M' 



^ptn{ct\, ' . •,«!) P?P(ah . . .,ak)+pn(ah • • -,^^-1) 



[22] 



/'2m(«2, . . .,«2) p7i{.ai, • . ■yak)-\-pn{.a2, . . .,«^-l) 



p2n+i(ah- ■ -.^i) Pn^jai, • • ■,2ai)—pJ(ah - • -,^3) 



/)2«+l(«2,. . .,«2) pn{<^2, . . .,2ai)—p„^(a2, . . .,a3) 



This gives in the following theorem : 



IV. Every number whose order is even may be expressed in 

 an unlimited number of different ivays as a quotient of sums, as 

 well as of differences, of squares of positive integers. 



The direct computation of these quotients for a given number 

 is for large values of the suffix very laborious, but recurrence- 

 formulae may be deduced by means of which, from the con- 

 stituents which enter into the quotients for a given suffix, the 

 quotient for the next higher suffix mav readily be computed. 

 Suppose that for some number of even order, 



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