104 



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



+ ai x"~2 — 32 x"-'* + as x"-' — an-i — 1 =^ 0, mod n, where 



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



32=:!. 2 + 1.3+ + 2.3+ 



33 = 1 .2.3 + 1 .2.4+ 



a„_i--l. 2.3 (n — 1). 



Hence, since the number of roots of a congruence with prime modulus 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. 



an— 1 == 1, mod n. 

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



Si — ai = 0. 



S2 — Si ai + 2a2 = 0. 



Sn-2 — S„_;^ai + — (n — 2) . a„_2==0. 



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



Sn — Sn— 1 . ai + + Si . an-1 = 0. 



Hence, 



Si^O, mod. n. San— 3 = mod. n. 



82 = 0, mod. n. S2n— 2 = — 1 mod. n. 



S2n— 1 — mod. n. 



Sn-2 = 0, mod. n. 



Sn— 1 ^ 1 mod. n. 

 Sn== 0, mod. n. 



or . 



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

 mod (n — 1). 



