190 History of the Theory of Numbers. [Chap, vii 
V. A. Lebesgue^^ noted that, given a primitive root g (g<p) of the 
prime p, we can find at once the primitive roots of p". Let g' be the positive 
residue <p~ when g^ is divided by p^ and set h = {g' — g)/p. Then 
g+px+p'^y {y = 0,. . ., p""^-!; x = 0,. . ., p-1; x^h) 
give p"~^(p — 1) primitive roots. Replacing g by g\ where i is less than 
and prime to p — 1, we obtain ]0(p") } primitive roots of p". In particular, 
a primitive root of p~ is a primitive root of p" (Jacobi^). But, if h = 0, g 
is not a primitive root of p". Since 
ginda+e^p_^ (mod p") , e = ip"-np-l), 
we can reduce by half the size of Jacobi's Canon. 
D. A. da Silva^^ gave two proofs that x'^ = l (mod p) has (f)(d) primitive 
roots, if d divides p — 1, and perfected the method of Poinsot^'^" for finding 
the primitive roots of a prime. 
F. Landry^^" was led to the same conclusion as Ivory.^^ In particular, 
if p = 2* + l, or if p = 2n+l (n an odd prime) and a7^p — l, any quadratic 
non-residue a of p is a primitive root. For each prime p< 10000, at least 
one prime ^ 19 is a quadratic non-residue of p. Cauchy's^* congruence for 
the primitive roots is derived and proved. 
G. Oltramare*^ proved that — 3°2^'' is a primitive root of the prime 
p = 2a/3 + l, if a^3, /3f^3, S'^^l, 22^^1 (mod p); that, if 
p = 3-2"'-M=g2+3r2, qx-]-ry = l, 
{ — l+qy — 3rx)5i^/2 is a primitive root of p; and analogous theorems. If 
a and 2a-^l are primes, 2 or a is a primitive root of 2a -fl, according as a 
is of the form 4n-[-l or 4n+3. If a is a prime 9^3 and if p = 2a4-l is a 
prime and m> 1, then 3 is a primitive root of p unless 3^'"~^-|-l=0 (mod p). 
[Cf. Smith.''^] 
P. Buttel^ attributed to Scheffler (Die unbestimmte Analytik, 1854, 
§142) the method of Crelle^^ for finding the residues of powers. 
C. G. Reuschle's^^ table C gives the Haupt-exponent {i. e., exponent to 
which the number belongs) (a) of 10, 2, 3, 5, 6, 7 with respect to all primes 
p< 1000, and the least primitive root of p; (b) of 10 and 2 for 1000< p< 5000 
and a convenient primitive root; (c) of 10 for 5000<p< 15000 (no primitive 
root given). Numerous errata have been listed by Cunningham."" 
Allegret^^ stated that if n is odd, n is not a primitive root of a prime 
2^^n-f-l, X>0; proof can be made as in Lebesgue.^^ 
"Comptes Rendus Paris, 39, 1854, 1069-71; same in Jour, de Math., 19, 1854, 334-6. 
**Proprietades geraes et resolu^ao directa das Congruencias binomias, Lisbon, 1854. Report 
by C. Alasia, Rivista di Fisica, Mat. e Sc. Nat., Pavia, 4, 1903, 25, 27-28; and Annaes 
Scientificos Acad. Polyt. do Porto, Coimbra, 4, 1909, 166-192. 
**'Troi8i&me m6moire sur la thdorie des nombres, Paris, 1854, 24 pp. 
"Jour, fiir Math., 49, 1855, 161-86. 
"Archiv Math. Phys., 26, 1856, 247. 
"Math. Abhandlung. . .Tabellen, Prog. Stuttgart, 1856; full title in the chapter on perfect 
numbers. i''^ 
"Nouv. Ann. Math., 16, 1857, 309-310. 
