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Full text of "History Of The Theory Of Numbers - I"

CHAP, ill]             FEKMAT'S AND WILSON'S THEOEEMS.                         73
A. Lista73 gave Lagrange's proof of Wilson's theorem.
V. Bouniakowsky74 gave Euler's22 proof.
P. L. Tchebychef75 concluded from format's theorem that
(z-l)(*-2). . .(s-p'+ij-af-M-lsO (mod p)
is an identity if p is a prime. Hence if Sj is the sum of the products of 1, . . . , p-1 taken j at a time, s?=0 0'<p-l), sp-i=-l (mod p), the last being Wilson's theorem.
Sir F. Pollock76 gave an incomplete statement and proof of the generalized Wilson theorem by use of associated numbers. Likewise futile was his attempt to extend Dirichlet's40 method [not cited] of association into pairs with the product = a (mod m) to the case of a composite m.
E. Desmarest77 gave Euler's13 proof of Fermat's theorem.
0. Schlomilch770 considered the quotient
-1)'+ (9 (n~2)p- • • •
J. J. Sylvester78 took x = 1, 2, . . . , p — 1 in turn in
where p is a prime. Since xp~l=l (mod p), there result p — l congruences linear and homogeneous in A^, . . ., Ap_2, ulp_i+l, the determinant of whose coefficients is the product of the differences of 1, 2, . . . , p — 1 and hence not divisible by p. Thus A^O,.:., 4p-i+lsO, the last giving Wilson's theorem.
W. Brennecke79 proved Euler's theorem by the methods of Horner37 and Laplace,23 noting that
(a'-^EEl (mod p2),          (ap-1)p2=l (mod p3), ....
He gave the proof by Tchebychef75 and his own proof.57
J. T. Graves80 employed nx=n+l (mod p), where p is a prime, and stated that, for n = l,..., p — l, then &=2,..., p in some order. Also x=p for n = p — l. Hence 2-3. . .(p — l)==p — 1 (mod p).
H. Dur&ge81 obtained (2) for a — x and Grunert's36 results on the series [m, n] by use of partial fractions for the reciprocal of x(x~l) . . .(x—ri).
E. Lottner82 employed for the same purpose infinite trigonometric and algebraic series, obtaining recursion formulae for the coefficients.
"Periodico Mensual Ciencias Mat. y Fis., Cadiz, 1, 1848, 63.
7*Bull. Ac. Sc. St. P6tersbourg, 6, 1848, 205.
74Theorie der Congrucnzen, 1849 (Russian); in German, 1889, §19.   Same proof by J. A.
Serret, Cours d'alg^bre supSrieure, ed. 2, 1854, 324. 78Proc. Roy. Soc. London, 5, 1851, 664. "TMorie des nombres, Paris, 1852, 223-5. 770 Jour, far Math., 44, 1852, 348.
"Cambridge and Dublin Math. Jour., 9, 1854, 84; Coll. Math. Papers, 2, 1908, 10. 79Einigc Satze aus den Anfangsgriinderi der Zahlenlehre, Progr. Realschuie Posen, 1855. "British Assoc. Report, 1856, 1-3. "Archiv Math. Phys., 30, 1858, 163-6. bid., 32, 1859, 111-5.