106 HlSTOEY OF THE THEORY OF NUMBERS. [CHAP. IV
Jean Plana6 developed \ (M - 1) +1 \ p and obtained
Take M=m, m- 1, . . . , 1 in the first equation and add. Thus
- . . .1
where 5,-= 1*4-2*+ . . . + (m — 1)*. For j> 1, we may replace p by j and get
a result obtained by Plana by a long discussion [Euler41]. He concluded erroneously that each s{ is divisible by m (for m = 3, s2 = 5).
F. Proth7 stated that, if p is a prime, 2p—2 is not divisible by p2 [error, see Meissner33].
M. A. Stern8 proved that, if p is an odd prime,
mp—m
,-,-p- • - •
for st- as by Plana and <T; = l*+2*+ . . . +m\ Proof is given of the formula below (2) of Eisenstein3 and Sylvester's formulae for q2 (corrected), as well as several related formulae.
L. Gegenbauer9 used Stern's congruences to prove that the coefficient of the highest power of re in a polynomial f(x) of degree p— 2 is congruent to (mp—m)/p modulo p if f(x) satisfies one of the systems of equations
E. Lucas10 proved that q2 is a square only for p = 2, 3, 7, and stated the result by Desmarest.4
F. Panizza11 enumerated the combinations p at a time of ap distinct things separated into p sets of a each, by counting for each r the combinations of the things belonging to r of the p sets:
(THCHDGD-ffl-
•Mem. Acad. Turin, (2), 20, 1863, 120.
7Comptes Rendus Paris, 83, 1876, 1288.
•Jour, fur Math., 100, 1887, 182-8.
"Sitzungsber. Ak. Wiss. Wien (Math.), 95, 1887, II, 616-7. 10Th£orie dea nombres, 1891, 423. "Periodico di Mat., 10, 1895, 14-16, 54-58.