IV. — On the Representation of Numbers as Quotients of Sums 

 and Differences of Perfect Squares 



BY ROBERT E. MORITZ 



Let the simple continuant^ whose elements ar, ^r+u • • • ^i-i, '^u 

 r<Ct, are positive integers, be denoted by 



p{ararJ{.\, . . . , at) or prj. 



The fundamental property of this function, which may be estab- 

 lished by induction, but most readily by expanding in terms of 

 the minors of the first s—r-\- 1 rows or columns its equivalent 

 determinant, is expressed by 



Pr,t= Pr,s ps-{-\,t-\-pr,s-l Ps^-ZJ, ^<i.t. [l] 



Limited by the definition of a continuant as ordinarily given, s be- 

 ing necessarily not less than r nor greater than /, pa,b is obviously 

 meaningless when a<ir, or <^>/, that is, when it involves elements 

 which do not exist in the continuant under consideration, and thus 

 no attempt seems to have been made to extend formula [i] to 

 values of ^<r or >/. Yet it frequently occurs that general expres- 

 sions based upon [i] are to be speciah'zed in a way which necessitates 

 5 to assume values outside the limits r and /, as when we expand a 

 continuant with reference to the ^th element, and then desire to let 

 /&=i or 2. Thus constituents like /i,o or ^2,-1 iTiay occur, which 

 vitiate the results and require that values of ^ which give rise to 

 such expressions be treated as separate cases. 



This is quite unnecessary. Since pa,b is meaningless as a con- 

 tinuant for a<Cr or (^>/, it may be given any meaning we please. 

 We shall therefore define it in a purely formal way by the funda- 

 mental equation [i]. 



iPor definition and fundamental theorems see Chrystal, Algebra, Part II, 

 p. 466. 



355 



