Classes of Congruent Integers. 



Now, to shew that the numbers (11) really represent a system 

 of bases, we have to prove, as in case I, 



/Iff IIe,=p- 1 , (14) 



where e b and e, c denote the exponents to which l+~"' and l+^ : 

 belong (mod. p"), and the products are to be taken for all values 

 of h and c. And then 



//(l +-"/"'. //(l+^) fl "'* !, (mod. p"), (15) 



f b f~n'i\}' '_i') excluding ths combination e 1 '=e/=... = e'j. +( j_ l =0. 



Making use of (10), we may replace (14) by 



i [>4"] +i [ n ~f b - ] + i [«=£■] =»-], when „>*, (16) 



or v(~/o^,-ii-l =7i-l, whenw^fc. (17) 



Now, it can easily be seen that if <* be a positive integer, 

 there are — ^- — — r positive integral values of x, which 



satisfy [%_4] -4 vi, *- [^] -1, [ p i-,] -2, [£] . 



But, among them there may be some multiples of p. The number 

 of such multiples is evidently equal to the number of multiples of p 



between — t^t and —p. excluding the former and including the 



latter. Hence it is equal to the number of positive integere 



between — T (exclusive) and — ^+r (inclusive), i.e. exactly 



L^J ~" Lp^J • Therefore there are 



( [^r] " [£] )"( tJr] - [^] ) «*»• ° f » («■ «>«'• 

 p.), which make log p -j- =*■• Hence 



J [«*f]-iM([^r]-[f])-(Cf]-c^]))- 



where p=\log p 7c], 



