208 
History of the Theory of Numbers. 
[Chap. VII 
use of the expansion of (1+2)^'^: 
M 
M-3 
Vl±8a = l±|2^a-— 7 2V=t— — 2V- . . . =tiV23"a"+ . 
N = 
M-3-5 
2-4 
(2n 
2-4-6 
■3) 
2-4-6-8. ..2n 
The coefficient of a" is an integer divisible by 2"^^ Retain only the terms 
whose coefficients are not divisible by 2'""^ and call their sum 6. Hence 
every term of 6~-^a is divisible by 2'". Thus the general solution of the 
proposed congruence is x=2'^~^x'^d. 
P. S. Laplace^*^" attempted to prove that, if p is a prime and p — l=ae, 
there exists an integer x<e such that x' — l is not divisible by p. For, if 
x = e and all earlier values of x make a:* — 1 divisible by p, 
/=(e^-l)-e^ 
would be divisible by p. 
,{e-lY-l\ + {^^y,{e-2Y-l\-... 
The sum of the second terms of the binomials is 
+ ... = -(1-1)^ = 0, 
while the sum of the first terms of the binomials is e ! by the theory of differ- 
ences, and is not divisible by p since e<p. [But the former equality implies 
that the last term of / is ( — 1)''(0— 1), whereas the theorem is trivial if x 
is allowed to take the value 0. Again, nothing in the proof given prevents 
a from being unity; then the statement that there is a positive integer 
x<p — l such that x^~^ — 1 is not divisible hyp contradicts Fermat's theorem.] 
L. Poinsot^^ deduced roots of a;"= 1 (mod p) from roots of unity. 
M. A. Stern^^ (p. 152) proved that if n is odd and p is a prime, rc"= —1 
(mod p) is solvable and the number of roots is the g. c. d. of n and p — l; 
while, if n is even, it is solvable if and only if the factor 2 occurs in p — 1 to a 
higher power than in n. 
G. Libri^^^ gave a long formula, involving sums of trigonometric func- 
tions, for the number of roots of x^+c=0 (mod p). 
V. A. Lebesgue^^ applied a theorem on/(a:i, . . ., Xk) = to derive Legen- 
dre's^^^ condition B^'=l for the existence of roots of (1), and the number 
of roots. Cf. Lebesgue^^ of Ch. VIII. 
Erlerus^^ (pp. 9-13) proved that, if pi, . . . , p^ are distinct odd primes, 
x~=l (mod2''p/'...p/) 
has 2", 2", 2"+^ or 2"+^ roots according as j/ = 0, 1, 2 or >2. 
For the last result and the like number of roots of x^=a, see the reports, 
in Ch. Ill on Fermat's theorem, of the papers by Brennecke^^ and Crelle^* 
of 1839, Crelle,^^ Poinsot" (erroneous) and Prouhet^^ of 1845, and Schering^''* 
of 1882. 
C. F. Arndt^^^ proved that the number of roots of x'= 1 (mod p") for 
"»Communication to Lacroix, Traitd Calcul Diff. Int., ed. 2, vol. in, 1818, 723. 
'"Jour, fur Math., 9, 1832, 175-7. See Libri," Ch. VIII. 
"*Archiv Math. Phys., 2, 1842, 10-14, 21-22. 
