Clisses of Congruent Integers. 5 



where the symbol {v T } denotes an integer which is divisible by p T , 

 but by no higher powers of p. By supposition d<zp — l, hence of 

 course d<;a(p—l) i.e. a+d<:ap. Therefore we get 



(l + 7r")P=l+{p«+*}. 

 Repeating the same reasoning, we get 



a+^f=a+{ V ^})»=i+{ V ^}, I (4) 



Hence if h be such an integer that a+Qi—1) d<zn£a + hd, then 

 <-'a = l> h - 



Let us now introduce the following notation: 



[:r] = 0, when.r^O, 



= rr, when x is a positive integer, ' 

 =the smallest integer greater than x, when x is positive 

 but not an integer. 



Then, we may write e a =pt d \ and consequently (2) may be 

 replaced by 



i ["-?] — >■ « 



Now, put n=\ ~ \d—d lt O^d^d, 



a = [-4-1 d - d,, é </,«?, 



then by subtraction, »— a—i -^- — -^- J d — ('/, — £?..). 

 Therefore, supposing «>«, we get 



