278 History of the Theory of Numbers. [Chap. DC 
R. D. Carmichael^^^ proved that, if a+1 and 2a + 1 are both primes, 
(a!)* — 1 is di^'isible by (a + l)(2a + l), and conversely. 
A. Ar^valo^^^ proved (6) and Lucas'" residues of binomial coefficients. 
N. G. W. H. Beeger"^ proved that [if p is a prime] 
(p_l)!+l = s-p+l (rnodp^), s = l+2^-'-\- . . .+{p-iy-' = pK.„ 
where h is a. Bernoulli number defined by the symbolical equation (/i + l)" 
= /i", hi = l/2. By use of Adams'^"" table of /i„ 2<114, it was verified 
that p = 5, p = 13 are the only p<114 for which (p — 1)!+1=0 (mod p^). 
T. E. Mason^^^ and J. M. Child^'^ noted that, if p is a prime >3, 
inp)\ = nl(piy (modp"+^). 
N. Nielsen^^'' proved that, if p = 2n+l, P=l-3-5. . . (2n-l), 
(-l)'^2np2=22'».3.5. . . (4n-l) (mod IGn^). 
If p is a prime >3, P=(-l)"2^"n! (mod p^). He gave the last result 
also elsewhere. ^"^^ 
C. I. Marks"- found the smallest integer x such that 2-4. . .{2n)x is di- 
visible by 3-5 ... (2n- 1). 
i»«Revista de la Sociedad Mat. Espanola, 2, "»Math. Quest. Educat. Times, 26, 1914, 19. 
1913, 130-1. ""Annali di mat., (3), 22, 1914, 81-2. 
"'Messenger Math., 43, 1913-4, 83-4. »«K. Danske Vidensk. Selsk. Skrifter, (7), 10 
""a Jour, fiir Math., 85, 1878, 269-72. 1913, 353. 
"STohoku Math. Jour., 5, 1914, 137. "'Math. Quest. Educ. Times, 21, 1912, 84-6. 
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