CHAPTER IV. 
RESIDUE OF (C/^->-l)/P MODULO P. 
N. H. Abel^ asked if there are primes p and integers a for which 
(1) a^-'=l (modp'), l<a<p. 
C. G. J. Jacobi^ noted that, for p^37, (1) holds only when p = ll, 
a = 3 or 9; p = 29, a = 14; p = 37, a = 18. Cf. Thibault^^ of Ch. VI. 
G. Eisenstein^ noted that, for p a prime, the function 
has the properties 
(2) quv=qu-\-qv, qu+pv^Qu-- (mod p), 
2g2=l-Ki-|+ • • • — ^^^\ Md p), 
P 1 o 
where s = (p+l)/2, . , ., p — 1. All solutions of (1) are included in a= 
u+puqu, 0<u<p. 
E. Desmarest^ noted that (1) holds for p = 4S7, a = 10, and stated that 
p = 3 and p = 487 are the only primes < 1000 for which 10 is a solution. 
J. J. Sylvester^ stated that, if p, r are distinct primes, p>2, then g^ 
is congruent modulo p to a sum of fractions with the successive denominators 
p — 1, . . . , 2, 1 and (as corrected) with numerators the repeated cycle of the 
positive integers ^r congruent modulo r to 1/p, 2/p,. . ., r/p. Thus, for 
r = 5, 
p — 1 p — 2 p — 6 p — 4: p — p — 
j,^^+.-l-+-i-+^+-^+-l-+... (p=10fc+7). 
p — 1 p—2 p — S p—4: p — 5 p — 
According as p = 4A;+l or 4A: — 1, ^2 is congruent to 
2.22.2. 2 
p—S p—4: p-7 p—S p — ll 
2 2 2 2 2 
p-2 p-3 p-Q p-7 p-10 " 
[the signs were given + erroneously]. For any p, 
?2= 7-\ ^ 5+... (modp). 
p—1 p—2 p—o 
iJour. fiir Math., 3, 1828, 212; Oeuvres, 1, 1881, 619. 
mid., 301-2; Werke, 6, 238-9; Canon Arithmeticus, Berlin, 1839, Introd., xxxiv. 
'BerUn Berichte, 1850, 41. 
*Th6orie des nombres, 1852, 295. 
'Comptes Rendus Paris, 52, 1861, 161, 212, 307, 817; Phil. Mag., 21, 1861, 136; Coll. Math. 
Papers, II, 229-235, 241, 262-3. 
105 
