274 ElSTOBY OP THE THEOBY OF NXJMBEBS. [CHAP. IX Glaisher94 discussed the residues modulo p3 of binomial coefficients. T. Hayashi95 proved that if p is a prime and fji+v=p, according as 0<s^, p<s<p, or s=0. T. Hayashi96 proved that, if 10 is the least positive residue of I modulo p, and if v=p—p>) modulo p. Special cases of the first result had been given by Lucas.86 A. Cunningham97 proved that, if p is a prime, (-l)- (mod p), B. Earn98 noted that, if (£), w=l,. . ., n— 1, have a common factor o>l, then a is a prime and n=ar. There is at most one prime <n which does not divide II (£) for m = l,. . ., n— 2, and then only when n+l=qar, where a is a prime and g<a. For m = Q, 1,. . ., n, the number of odd (£) is always a power of 2. P. Bachmann" proved that, if h(p~ 1) is the greatest multiple <k of - (mod p)- the case Jfc odd being due to Hermite.76 G. Fontene" stated and L. Grosschmid100 proved that -i)s(-1}t (modp)- p=p A. Fleck101 proved that, if Ogp<p, a+b=0 (mod p), N. Nielsen102 proved Bachmann^s" result by use of Bernoulli numbers. wQuar. Jour. Math., 31, 1900, 110-124. MJour. of the Physics School in Tokio, 10, 1901, 391-2; Abh. Geschichte Math. Wiss., 28, 1910, 26-28. MArchiv Math. Phys., (3), 5, 1903, 67-9. "Math. Quest. Educat. Times, (2), 12, 1907, 94r-5. 98Jour. of the Indian Math. Club, Madras, 1, 1909, 39-43. "Niedere Zahlentheorie, II, 1910, 46. 100Nouv. Ann. Math., (4), 13, 1913, 521-1. 1MSitzungs. Berlin Math. GeselL, 13, 1913-4, 2-6. Cf. H. Kapferer, Archiv Math. Phys (3), 23, 1915, 122. l«Annali di mat., (3), 22, 1914, 253.