PROCEEDINGS 



OF THE 



The Direct Solution of the Quadratic and Cubic Binomial 

 Congruences with Prime Moduli. By H. C. PocKLiNGTON, M.A., 

 St John's College. 



[Received 22 January 1917: read 5 February 1917.] 



1. The solution of congruences by exclusion methods, 

 although easy enough when the modulus is moderately large, 

 becomes impracticable for large moduli because the labour varies 

 as the modulus or its square root. In a direct method the labour 

 varies roughly as the cube of the number of digits in the modulus, 

 and so remains moderate for large moduli. The object of this 

 paper is to develop the direct method. We take or = a, mod. p, 

 first, discussing the cases where p = 4<m + 3 and p = Sin + 5 in 

 § 2 and that where p = 8ni + 1 in § 3. We next take ar^ = a and 

 discuss the cases where p = Sm + 2, /> = 9m + 4 and p=i9m + 7 

 in § 4 and that where p = dm + 1 in § 5. 



2. Throughout the paper we suppose the modulus to be 

 p where p is prime*. If p is of the form 4wi + 3 the solution of 

 ^•- = a is X = ± a'^'^\ If p is of the form 8m + 5 the solution is 

 ^ = ± a'"+i provided that a-'^+^ = 1. But if not, a'^"*+i = - 1, and 

 as 2 is a non-residue 4-'"+' b - 1 ; so that (4a)-»*+^ = 1 and we have 

 2/ = + (4a)'"+^ as the .solution of y- = 4o. Hence 



x=±y;2 or x=±{p+ y)/2 



is the solution of x-=a. These values of x can be calculated 

 without serious difficulty by repeated squaring (followed by division 

 by the modulus to find the remainder) and multiplication of the 

 numbers so found (again followed by division). 



* Hence if it is composite we must factorize it and solve the congrnence for 

 each of the different prime factors. 



VOL. XIX. PARTS II., III. 



