188 
History of the Theory of Numbers. 
[Chap. VII 
E. Prouhet^" gave, without reference, Crelle's^' method of forming the 
residues of the powers of a number. The object of the paper is to give a 
uniform method of proof of theorems, given in various places in Legendre's 
text, relating to the residues of the first n powers of an integer belonging 
to the exponent n modulo P, especially when P is a prime or a power of a 
prime, and the existence of primitive roots. He gave (p. 658) the usual 
proof that =•= 2 is a primitive root of a prime 2^+ 1 if 5 is a prime 4/c± 1 (with 
a misprint). 
C. F. Arndt^^ proved that if ^ is a primitive root of the odd prime p and 
if p^ (SKn) is the highest power of p dividing G = g^~^ — 1, then g belongs 
to the exponent p'*~''(p — 1) modulo p". Conversely, if the last is true of a 
primitive root g of p, then G is divisible by p^ and not by p^"'"^ The first 
result with X = 1 shows that any primitive root of p^ is a primitive root of 
p", n>2. Let g he a, primitive root of p; if G is not divisible by p^, g is a. 
primitive root of p^; but if G is divisible by p^, and h is not divisible by p, 
then g+hp is a primitive root of p^. Any odd primitive root of p** is a 
primitive root of 2p". If gr is a primitive root of p'* or 2p'*, and t is a divisor 
of p"~^(p — 1), then if a ranges over the integers <t and prime to t, the 
<f>{t) integers belonging to the exponent t modulo p" or 2p" are g% where 
e = p"~^(p — l)a/<. The numbers belonging to the exponent 2"""* modulo 
2" are found more simply than by Gauss'^ and Jacobi^^ (p. 37). 
P. L. Tchebychef^^ proved that if a, /3, . . . are the distinct prime factors 
of p — 1, where p is a prime, then a is a primitive root of p if and only if no 
one of the congruences x'' = a, xP = a,. . . (mod p) has an integral root. 
This furnishes a method (usually impracticable) of finding all primitive 
roots of p. A second method uses a number a belonging to the exponent n, 
and a number h not congruent to a power of a, and deduces a number 
belonging to an exponent >n. In the second supplement, he proved that 
3 is a primitive root of any prime 2^"+ 1 ; that =*= 2 is a primitive root of any 
prime 2a +1 such that a is a prime 4A:± 1 ; 3 is a primitive root of 4iV2"'4-l 
if w>0 and iV is a prime >9^ /(4-2'"); 2 is a primitive root of any prime 
4iVH-l such that A^ is an odd prune. The last result was later proposed'^ 
as a question for solution (with reference to this text) . There is given the 
table of primitive roots and indices for primes < 200, due to Ostrogradsky^^. 
Schapira (p. 314) noted that in the list of errata in Jacobi's^^ Canon (p. 222) 
there is omitted the error 8 for 6 in ind 14 for p = 25. 
V. A. Lebesgue^*^ remarked that Cauchy's^^ congruence X=0 shows 
the existence of 0(n) integers belonging to the exponent n modulo p, a 
prime. 
»«Nouv. Ann. Math., 5, 1846, 175-87, 659-62, 675-83. 
»3Jour. fiir Math., 31, 1846, 259-68. 
"Theory of Congruences (m Russian), 1849. German translation by Schapira, Berlin, 1889, 
p. 192. Italian translation by Mile. Massarini, Rome, 1895, with an extension of the 
tables of indices to 353. 
»Nouv. Ann. Math., 15, 1856, 353. Solved by use of Euler's criterion by P. H. Rochette, 
and., 16, 1857, 159. Also proved by Desmarest,*^ p. 278. 
»*Nouv. Ann. Math., 8, 1849, 352; 11, 1852, 420. 
