( 658 ) 



before found. Thej also seem to be verj rare in the German diluvium. 

 As far as I iiave been able to fujd out, they arc made mention of 

 by Stollky ^) only. 



Mathematics. — "An onnh/ticaJ e.c})r('sswn for the (j rentest common 

 divisor of tiro uile<iers.'' By Prof. J. C. Kluyvek. 



111 this papoi' Ave jn'opose to construct certain functions z of two 

 real vai-iablcs ,r and y whicli for positive and integer values of tlicse 

 variables become e<pial to the greatest common divisor of ,/■ and y. 



A very simj)le sobition of this })rol)lcm is obtained as follows. 

 Denote by [^/] the integer part of the number u and consider the 

 arithmetical discontinuous function 



Fi^„):=n-\.\-^. 

 For auy iutegci' n \vc iuive 



r{n + H) = r[u), p{u -f 0) =z - i-, r{„ -O)^^ ^. 



We will take 



an<l cousctpicntly 



[n] — [n — 0] — n — \. 



Integer vahics of n excepted tiie wcllknown relation 



"='=^ sin 2 Ji nil 



holds and from it we deduce the idejitical erpiation 



''~k r (n + — I =~i P (u -f — I = P (n h), 



where a and ,i arc relative pi-ime integers. That iIm^ identity is 

 still valid for integer values of ii. may l»e easily verified. 

 By the equation 



. = i^'p('^\ (I) 



a discontinuous function of the variables .'■ and // is dclined. We 

 may regard it as a tii-st solution of the [)ro[)osed problem. For if 



^) Stolley. loc. cit. p. 41. 



