Chap. IV] RESIDUE OF (U^ ^ — l)/p MODULO p. 109 
Glaisher^^ considered Qu in connection with BernouUian numbers and 
gave 
^-^^-i(}+l+ +1) (modp = 3ft+l). 
A. Pleskot^^ duplicated the work of Plana. ^ 
P. Bachmann^° gave an exposition of the work by Sylvester,^ Stern,* 
Mirimanoff.-^^ 
M. Lerch^^ set, for any odd integer p and for u prime to p, 
P 
Then,* as a generahzation of (2), 
qnv =qu+qv, Qu+pv =qu-\ (mod p) , 
"-I^H- 2g.-2i— 2I (mod p), 
where v ranges over the positive integers <p and prime to p; X over those 
>p/2; fi over those <p/2. Henceforth, let p be an odd prime and set 
N=\{p-l)\+l\/p. Then N^q,+ . . .+q,.^, 
[P/4]1 [p/3]i [p/5]i [2p/5U 
32^-1 si, 3^3= -2 si, 5^5^-2 2^-2 Si 
»=l^ v=l^ a=l« 5=1 
modulo p. If \p(n) is the number of sets of positive solutions <p oi ixv = n 
and hence the number of divisors between n/p and p of n, 
Employing Legendre's symbol and BernoulHan numbers, we have 
^= sY^)g =0 or (-1)"-^2j5„ (mod p), 
v=l \p/ 
according as p = 4n+3 or 4n+l. In the respective cases, 
p-i 
\{^^Vq.^Cl{-p)oTO{modp), 
where CZ(— A) is the number of classes of positive primitive forms 
ax^+bxy+cy^ of negative discriminant 6^— 4ac= —A. Also, modulo p, 
y^iv'^LpJ' ^ ^aaLpJ' 
6 ao Lp J a aabf-pj 
where a, a are quadratic residues of p, and 6, /3 non-residues. 
isProc. London Math. Soc, 33, 1900-1, 49-50. 
"Zeitschrift fiir das Realschulwesen, Wien, 27, 1902, 471-2. 
"Niedere Zahlentheorie, I, 1902, 159-169. *The greatest integer ^x is denoted by [x]. 
"Math. Annalen, 60, 1905, 471-490. 
