274 History of the Theory of Numbers. [Chap, ix 
Glaisher^^ discussed the residues modulo p^ of binomial coefficients. 
T. Hayashi^^ proved that if p is a prime and fjL+v = p, 
(nsr>(-i)'C).».i(-<ip)- 
according as 0<s^v, v<s<p, or s = 0. 
T. Hayashi^^ proved that, if Iq is the least positive residue of I modulo p, 
and if v = p—ii, 
modulo p. Special cases of the first result had been given by Lucas. *^ 
A. Cunningham^^ proved that, if p is a prime, 
(^;^)^(-ir (modp), ^(p^)^! (modp^p>3). 
B. Ram^^ noted that, if (^), m = l,. . ., n — 1, have a common factor 
o>l, then a is a prime and n = a''. There is at most one prime <n which 
does not di\dde n(^) for m = l,. . ., n — 2, and then only when n+l=?a^ 
where a is a prime and q<a. For m = 0, 1, . . ., n, the number of odd (^) 
is always a power of 2. 
P. Bachmann^^ proved that, if h{p — l) is the greatest multiple <A; of 
p-i, 
(,!i)+(2(pii))+-+(M/-i>''("^°'^^>' 
the case k odd being due to Hermite.'^ 
G. Fonten^ stated and L. Grosschniid^°° proved that 
(p(pil))^(-l)' (^odp), P = p\ a^O. 
A. Fleck^^i proved that, if 0^p<p, aH-6=0 (mod p), 
N. Nielsen^"^ proved Bachmann's^^ result by use of Bernoulli numbers. 
wQuar. Jour. Math., 31, 1900, 110-124. 
"Jour, of the Physics School in Tokio, 10, 1901, 391-2; Abh. Geschichte Math. Wias., 28, 1910, 
26-28. 
"Archiv Math. Phys., (3), 5, 1903, 67-9. 
•'Math. Quest. Educat. Times, (2), 12, 1907, 94-5. 
"Jour, of the Indian Math. Club, Madras, 1, 1909, 39-43. 
"Niedere Zahlentheorie, II, 1910, 46. 
""•Xouv. Ann. Math., (4), 13, 1913, 521-4. 
"'Sitzungs. BerUn Math. Gesell., 13, 1913-4, 2-6. Cf. H. Kapferer, Archiv Math. Phys. 
(3), 23, 1915, 122. 
"«Annali di mat., (3), 22, 1914, 253. 
