History Of The Theory Of Numbers - I

CHAP. IX] DlVISIBILTY OF FACTOKIALS, MULTINOMIAL COEFFICIENTS. 277
...p1, P/=p'+1...pp-1 with * = (p-l)/2, when p=4fc-l; but P=ppp~1 pV-3. . ., P'=py-2 pV-4. . ., when p=4fc+l.
G. Oltramare130 gave several algebraic series for the reciprocal of the binomial coefficient (2^) and concluded that, if the moduli are primes,
V. Bouniakowsky131 considered the integers qi,. .., qa, each <2V and prune to N, arranged in ascending order of magnitude. If X is any chosen integer ^s, multiply
qa=N-ql, fc-i^AT-fe,. - -, &-x+i=#-&
together and multiply the resulting equation by QI. . .q*-\. Apply the generalized Wilson theorem q^. . .&+(— 1)*=0 (mod JV). Hence
ffi fc • • • &• ff i ft - • - ff.-x+ ( - l)5+x= 0 (mod N) . For N a prime, we have s=AT— 1 and
X!(Ar-l-X)!+(-l)x=0 (mod TV) (l^X^tf-l).
C. A. Laisant and E. Beaujeux132 gave the last result and
F. G. Teixeira133 proved that if a = 22p"lp-a) a<2p-l,
(mod a+a+l+o+2+...+a+2p-l). Thus, for p = 3, a = 1, a = 95,
95-96-97-98-99-100=32-52-3 (mod 585-95+... +100).
M. Vecchi134 noted that the final formula by Bouniakowsky131 follows by induction. Taking \ = (N—1)/2, we get Lagrange's formula (4). From the latter, we get
( — 1)~ (mod p).
The case y = (p-l)/2 gives Arndt's118 result
P+I (6) {3-5-7...(p-2)}2EE(~l)~
Vecchi135 proved that, if v is the number of odd quadratic non-residues of a prime p = 4n+3, then 1-3-5... (p—2)=( — 1)" (mod p). If JJL is the number of non-residues <p/2, 1-3-5. , . (p-2)=(-l)M+12(p-1)/2 (modp).
130M6m. de Tlnstitut Nat. Genevois, 4, 1856, 133Jornal de Sciencias Math, e Astr., 3, 1881,
33-6. 105-115.
131Bull. Ac. Sc. St. Ptesbourp, 15, 1857,202-5. 134Periodico di Mat., 16, 1901, 22-4.
132Nouv. Corresp. Math., 5, 1879, 156 (177). ™Ibid., 22, 1907, 285-8.