Chap. VII] PRIMITIVE RoOTS, EXPONENTS, INDICES. 185 
and q are primes, 2 and —2 are primitive roots of p. If p = 4g+l and 
g = 3n+l are primes, 3 and —3 are primitive roots of p. 
F. Minding^^ gave without reference Gauss' second proof of the exist- 
ence of primitive roots of a prime. 
F. J. Richelot^^ proved that, if p = 2"'+l is a prime, every quadratic 
non-residue (in particular, 3) is a primitive root of p. 
A. L. Crelle^^ gave a table showing all prime numbers ^ 101 having a 
given primitive root; also a table of the residues of the powers of the 
natural numbers when divided by the primes 3, . . ., 101. His device^^ 
for finding the residues modulo p of the powers of a will be clear from the 
example p = 7, a = 3. Write under the natural numbers <7 the residues 
of the successive multiples of 3 formed by successive additions of 3 ; we get 
12 3 4 5 6 
3 6 2 5 14. 
Then the residues 3, 2, 6, . . . of 3, 3^, 3^, . . . modulo 7 are found as follows: 
after 3 comes the number 2 below 3 in the above table; after 2 comes the 
number 6 below 2 in the table; etc. 
Crelle^° proved that, if p is a prime and X is prime to p — 1 and <p — 1, 
the residues modulo p of z^ range with z over the integers 1, 2,. . ., p — 1. 
His proof that there exist (^(n) numbers belonging to the exponent n 
modulo p, if n divides p — 1, is like that by Legendre.^ 
G. L. Dirichlet^^ employed 0(A:) systems of indices for a modulus 
/j = 2^p'p"'. . ., where p, p', . . . are distinct primes, and X^3. Given any 
integer n prime to k, and primitive roots c, c', . • • of P'> v" \ • • • > we can 
determine indices a, /3, 7, 7', . . . such that 
n=(-l)''5^ (mod 2^^), n = c^ (mod p'), n = c"'' (mod p"'),- • •• 
Michel Ostrogradsky^^ gave for each prime p<200 all the primitive 
roots of p and companion tables of the indices and corresponding numbers. 
(See Jacobins and Tchebychef .3^) 
C. G. J. Jacobi^^ gave for each prime and power of a prime < 1000 two 
companion tables showing the numbers with given indices and the index 
of each given number. In the introduction, he reproduced the table by 
Burckhardt, 1817, of the length of the period of the decimal fraction for 
1/p, for each prime p^2543, and 22 higher primes. Of the 365 primes 
<2500, we therefore have 148 having 10 as a primitive root, and 73 of the 
form 4w+3 having —10 as a primitive root. Use is made also of the 
primes for which 10 or — 10 is the square or cube of a primitive root. 
"Anfangsgrlinde der hoheren Arith., 1832, 36-37. 
"Jour, ftir Math., 9, 1832, p. 5. 
^Hhid., 27-53. 
"Also, ihid., 28, 1844, 166. 
"Abh. Ak. Wiss. BerUn, 1832, Math., p. 57, p. 65. 
^HhU., 1837, Math., 45; Werke, 1, 1889, 333. 
"Lectures on alg. and transc. analysis, I-II, St. P^tersbourg, 1837; M6m. Ac. Sc. St. P6tera- 
bourg, s6r. 6, sc. math, et phys., 1, 1838, 359-85. 
2'Canon Arithmeticus, Berlin, 1839, xl+248 pp. Errata, Cunningham.""-"" 
