CHAP, vni] SUM OP FOUR SQUARES. 287 



Unless = /?,' = /3', we may set & = a + 4A, ' = ' + 4A, A > 0, if 

 the term be repeated. Thus 



6/3'] 

 + 2#[>(a + 6) + 46A] + 2N[a'(a + 6) + 46A]. 



Let c range over both the a and a' numbers. Then 



/* = tf[c(a + 6)] + 2N[c(a + 6) + 46A] - 2tf[aa + 6/3']. 



In the second term set c = d -f 4A#, d < 4A, B ^ 0. Now a + 6 may 

 represent any even number 2(7, and 6 + B(a -f- 6) any odd number e. 

 Thus 



/z = tf [c(o + 6)] + 2JV[2Cd + 4Ae] - 2N[aa + 6/3']. 



Since a + /3' = (mod 4), a 4= 6. Thus the second member of 



2N[aa: + 6/3'] = N[aa + 6/3'] + AT[a/3' + 6a] 



is twice the like sum with 6 > a. Set 6 = a + 2G, a + /3' = 4A. Then 

 N[> a + 6/3'] = AT[2/3'(? + 4Aa] + A^[2aG + 4Aa] = N[2dG + 4Ao], 



where d < 4A. Hence ^ = A^[c(a + 6)]. Here c ranges over all the 

 divisors of p. If p = cf, the equation 2p = c(a + 6) becomes 2/ = a + 6, 

 which has / sets of odd solutions. But Zp/c is the sum of the divisors of p. 

 Thus IJL = <r(p). 



T. Schonemann 31 used the notation cos n, sin n for a pair of solutions 

 of x 2 + y 2 = 1 (mod p). If cos m, sin m is the notation for a second pair 

 of solutions, then the expansions of cos (n + m), sin (n + m) give a third 

 pair of solutions. Then, for a an integer, 



(cos n + i sin n) a = cos an + * sin cm (mod p). 



If p is a prime, cos pn = cos n, sin pn = ( l)^- 1 )/ 2 sin n (mod p). Hence 

 cos (p ^F l)rz, = 1 if p = 4k =t 1. An integer a is put into " class A " 

 if 1 a 2 is a quadratic residue of p, otherwise into class B. It is proved 

 that if cos n belongs to class A and if a is the least integer for which 

 cos an = 1 (mod p), then a is a divisor of p =F 1 when p = 4k 1; then 

 cos n is said to belong to the number a. There exist <f>(p 1) " primitive ' 

 cosines which belong to p 1. For p = 4n -f- 1, cosn is primitive, so 

 that all sets of real solutions of x 1 + y 2 = 1 (mod p) are given by cos in, 

 sintn for t = 1, 2, , p 1; the cases of coincidence are found. The 

 result is that for any prime 8m 1, 8m + 3 or 8m + 5, there are m essen- 

 tially different sets of solutions, provided O 2 + I 2 = 1 is excluded. The 

 same ideas are applied to the determination of the quadratic character 

 of 2, 3, 5. 



G. Eisenstein 32 stated without proof that the number of all repre- 

 sentations of an odd integer m as a 3] is 80- (m) [Jacobi 24 ], and that, if 



31 Jour, fur Math., 19, 1839, 93-110. 



32 Jour, fur Math., 35, 1847, 133; Math. Abhandlungen, 1847, 193. In Jour, de Math., 17, 



1852, 477, the first result is said to follow from a property of ternary quadratic forms. 



