Classes of Congruent Integers. 15 



=(i+(io,£K+ f 



=(i+{p+(i^i) !, -'^' , - 1 v}(i^K'+ ) v " 



= i + l ( p+(^cj i y-'-' x "- 1 ' l Y^cj i )p^'-+ 



== l + jp*+a+rt ! i+r 2 \ x ^ (29) 



But if at least one of the exponents be not divisible by p, 



//( 1 + ?,-•/« = l + (p J }. 



On account of these ambiguities a^O, x.^0, the reasonings in the 

 preceding cases cannot be applied to the present case in general. 



If, however, néd + h, theri, whatever be the values of x 1 and 

 x 2 , we get always 



i ? (mod. p") 



and 77(1 + f„7r*) e ki = 1 or 1 + [f] 



according as the exponents e' ki are all divisible by p or not. 

 Hence we can proceed in the same manner and arrive at the same 

 conclusion as in the last paragraph. 



In the following we shall consider the case n>ä+le. From 

 (29), we see that ar 2 > or =0, according as p + {2c k gl) p ~ lTci is or is 

 not divisible by a power of p higher than the d' A . Hence we 

 distinguish two cases, according as the congruence 



^+7rV" 1 =0 J (mod. p' m ), (30) 



has a solution or not. If we determine an integer P from 



p = tt'7>, (mod. p' i+1 ), 



we may replace (30) by 



p+i H =0, (mod. p). (31) 



If (31) has no solution, then .r,=u\, = 0; and following exactly the 

 same reasoning as in the last paragraph, we can shew that the 



