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

CHAP. IV]                 RESIDUE OF (Up-l--l)/p MODULO p.                      109
Glaisher18 considered qu in connection with Bernoullian numbers and gave
A. Pleskot19 duplicated the work of Plana.6
P. Bachmann20 gave an exposition of the work by Sylvester,8 Stern,8 Mirimanoff.12
M. Lerch21 set, for any odd integer p and for u prime to p,
flu-(u*W-l).
Then,* as a generalization of (2),
(mod p),
^ M?]'     2*H1S -s (mod p)'
where j> ranges over the positive integers <p and prime to p; X over those >p/2; /A over those <p/2.   Henceforth, let p be an odd prime and set
N=\(p-l)l+l\/p.   Then #=<?!+... +<?P-!,
[p/4]l                                    (p/3h                                     [p/53-.         [2p/5H
<?2=-*S'i,        3g3=-2S*        5g5=-2S-2Si
ys-l*'                                     vs"!*7                                     a-l1*         Z>=1
modulo p.   If \l/(ri) is the number of sets of positive solutions <p of /zi>=n and hence the number of divisors between n/p and p of n,
Employing Legendre's symbol and Beraoullian nxunbers, we have
j4=5Xp)9'" r (-W23" (mod P) according as p = 4n+3 or 4n+l.   In the respective cases,
,ssCZ(--p) or 0 (mod p),
where  CZ(~^)  is  the  number of classes of positive primitive forms ax2+l)xy-{-cy2 of negative discriminant b2 4ac= A.   Also, modulo p,
where a, a are quadratic residues of p} and 6, ft non-residues.
c. London Math. Soc., 33, 1900-1, 49-50. lflZeitschrift fUr 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.