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208                         HlSTOEY OF THE  THEORY OF NlJMBEBS.              [CHAP. VII
use of the expansion of
M-3-5...(2n-3) 2-4-6-8...2n
The coefficient of an is an integer divisible by 2n+1. Retain only the terms whose coefficients are not divisible by 2m~1 and call their sum 6. Hence every term of 02+a is divisible by 2m. Thus the general solution of the proposed congruence is z=2m~ V=*=0.
P. S. Laplace160 attempted to prove that, if p is a prime and p — 1 = ae, there exists an integer x<e such that xe— 1 is not divisible by p. For, if s=e and all earlier values of x make xe— I divisible by p,
would be divisible by p.   The sum of the second terms of the binomials is
while the sum of the first terms of the binomials is e! by the theory of differences, and is not divisible by p since e< p. [But the former equality implies that the last term of / is ( — l)e(0— 1), whereas the theorem is trivial if x is allowed to take the value 0. Again, nothing hi the proof given prevents a from being unity; then the statement that there is a positive integer x< p — l such that xp~l — 1 is not divisible byp contradicts Fermat's theorem.]
L. Poinsot11 deduced roots of 2n= 1 (mod p) from roots of unity.
M. A. Stern15 (p. 152) proved that if n is odd and p is a prime, zn= — 1 (mod p) is solvable and the number of roots is the g. c. d. of n and p — 1 ; while, if n is even, it is solvable if and only if the factor 2 occurs in p — 1 to a higher power than hi n.
G. Libri161 gave a long formula, involving sums of trigonometric functions, for the number of roots of z2+c=0 (mod p).
V. A. Lebesgue13 applied a theorem on/0&i> • • •> #&)— 0 to derive Legen-dre's154 condition Bp'=l for the existence of roots of (1), and the number of roots. Cf. Lebesgue17 of Ch. VIII.
Erlerus25 (pp. 9-13) proved that, if PI, . . ., pM are distinct odd primes, afel (mod2'p1-»...p/)
has 2*, 2", 2*+1 or 2"+2 roots according as i> = 0, 1, 2 or >2.
For the last result and the like number of roots of ofea, see the reports, in Ch. Ill on Fermat's theorem, of the papers by Brennecke57 and Crelie58 of 1839, Crelie,66 Poinsot67 (erroneous) and Prouhet69 of 1845, and Sobering102 of 1882.
C. F. Arndt162 proved that the number of roots of ofe 1 (mod pn) for
"Communication to Lacroix, Trait£ Calcul Diff. Int., ed. 2, vol. in, 1818, 723. wlJour. fur Math., 9, 1832, 175-7.    See Libri," Ch. VIII. 1MAxchiv Math. Phys., 2, 1842, 10-14, 21-22.