104 



But the congruence is of the degree n — 2, since it may be written 



+ ai x"— ^ — 32 x"-^^ + as x"-* — an— 1 — 1 = 0, mod n, where 



ai = l + 2 + 3+ +(n-l) 



32 = 1.2 + 1.3+ + 2.3+ 



33 = 1 .2.3 + 1 .2.4+ 



an-i^l. 2.3 (n — 1). 



Hence, since the number of roots of a congruence with prime modnlus can 

 not be greater than the modulus, the given congruence must be identical. Hence, 



ai = 0, mod n. 



a2 ^ 0, mod n. 



an— 2 = 0, mod n. 



En-i = 1, mod n. 

 But from the theory of symmetric functions we have the following relations: 



Si — ai = 0. 



Ss — Si ai + 2a2 = 0. 



Sn-2 — S„-3ai + — (n — 2) . an-2=0. 



Sn-l — Sn-2 . ai+ +(n — 1) . an-l = 0. 



Sn — Sn-l . 31 + + Si . an-1 ^ 0. 



Hence, 



Si = 0, mod. n. Szn— 3^0 mod. n. 

 S2 = 0, mod. n. S2n— 2 = — 1 mod. n. 

 S2n-i — mod. n. 



or 



Sn-2 =0, mod. n. 

 Sn-l = 1 mod. n 

 Sn^ 0, mod. n. 



Sk = 0, mod n, when k S mod (n — 1 ) and Sk = — 1 , mod n, when k : 



mod (n — 1). 



