106 History of the Theory of Numbers. [Chap, iv 
Jean Plana^ developed j (Af — 1) + 1 } " and obtained 
M''-M-\{M-iy-{M-l)\=pf{M), 
Take M = m, 7n — l,. . ., 1 m the first equation and add. Thus 
=/(!) + . . . 4-/(w) =Si+^^S2-f • • • +Sp-i, 
where s, = r+2'H- . . . + (m — 1)'. For j> 1, we may replace p by j and get 
m 
-w=is,_i+ (0s,_2+ Qjsi-aH- • • • +isi, 
a result obtained by Plana by a long discussion [Euler"]. He concluded 
erroneously that each Si is divisible by m (for m = 3, S2 = 5). 
F. Proth^ stated that, if p is a prime, 2''— 2 is not divisible by p^ [error, 
see Meissner^^]. 
M. A. Stern^ proved that, if p is an odd prime, 
rrf — m_ , , , 1 _ , i , ,1 
— - — =Si-§S2+iS3-. . .-— YSp-i=o-p-i+io-p_2+. . • + —^0-1 
for Si as by Plana and 0-^ = l'+2*+ • • . +w\ Proof is given of the formula 
below (2) of Eisenstein^ and Sylvester's formulae for q2 (corrected), as well 
as several related formulae. 
L. Gegenbauer^ used Stern's congruences to prove that the coefficient of 
the highest power of a^ in a polynomial f{x) of degree p — 2 is congruent to 
{m^—m)/p modulo p if /(a:) satisfies one of the systems of equations 
J{\) = {-lf^'V-\{m-\), /(X)=X--VxW (X = l,. . ., p-1). 
E. Lucas^° proved that ^2 is a square only for p = 2, 3, 7, and stated the 
result by Desmarest.^ 
F. Panizza^^ enumerated the combinations p at a time of ap distinct 
things separated into p sets of a each, by counting for each r the combina- 
tions of the things belonging to r of the p sets : 
(T)=i.(^>a) CO ■•(")' 
•Mem. Acad. Turin, (2), 20, 1863, 120. 
^Comptes Rendus Paris, 83, 1876, 1288. 
•Jour, fur Math., 100, 1887, 182-8. 
•Sitzungsber. Ak. Wiss. Wien (Math.), 95, 1887, II, 616-7. 
">Th6orie des nombres, 1891, 423. 
"Periodico di Mat., 10, 1895, 14-16, 54-58. 
