272 History of the Theory of Numbers. [Chap, ix 
J. Wolstenholme"^ noted that the highest power of 2 dividing i^""^^) 
isq — p — l, where q is the sum of the digits of 2m — I to base 2, and 2" is the 
highest power of 2 dividing ?«. 
J. Petersen"^ proved by Legendre's formula that C^'') equals the 
product of the powers of all primes p, the exponent of p being (ta+tb — ta+b) 
-^(p — 1), where ta is the sum of the digits of a to base p. 
E. Cesaro^° treated Kummer's^^ problem. He stated (Ex. 295) and 
Van den Broeek^^ proved that the exponent of the highest power of the 
prime p dividing (-„") is the number of odd integers among [2n/p], [2n/p^], 
[2n/p'],.... 
O. Schlomilch^^" stated in effect that („ + i) is divisible by n. 
E. Catalan^'- proved that if n is odd, 
p:)+io(t-^) 
= (modn+2). 
W. J. C. Sharp^^" noted that {p-\-n)\ — p\n\ is divisible by p^, if p is a 
prime >n. This follows also from (''t") — 1 (mod p) [Dickson^"]. 
L. Gegenbauer^^ noted that, if a is any integer, r one of the form 6s or 
3s according as n is odd or even, 
The case n odd, a = 2, r = 3, gives Catalan's result. 
E. Catalan^ proved Hermite's'^^ theorem. 
Ch. Hermite^^ stated that (Z) is divisible by m— n-f-1 if w is divisible 
by n; by (m— n+l)/€ if e is the g. c. d. of m+1 and n; by m/8, if 5 is 
the g. c. d. of m, n. 
E. Lucas^^ noted that, ifn^p — 1, p — 2, p — 3, respectively, 
(^;3)-(-ir(^^±lM)(:nodp), 
if p is a prime, and proved Hermite's'^ result (p. 506). 
F. RogeP^ proved Hermite's"^ theorem by use of Fermat's. 
^*Jour. de math. 6\6m. et spec., 1877-81, ex. 360. 
^»Tidsskrift for Math., (4), 6, 1882, 138-143. 
soMathesis, 4, 1884, 109-110. 
8'7feid., 6, 1886, 179. 
«>"Zeitschrift Math. Naturw. Unterricht, 17, 1886, 281. 
<«M6m. Soc. Roy. Sc. de Li6ge, (2), 13, 1886, 237-241 ( = Melanges Math.). Mathesis, 10, 1890. 
257-8. 
82aMath. Quest. Educ. Times, 49, 1888, 74. 
s^Sitzunpsber. Ak. Wiss. Wien (Math.), 98, 1889, Ila, 672. 
wM6m. Soc. Sc. Li^KC, (2), 15, 1888, 253-4 (Melanges Math. III). 
«*Jour. de math, sp^ciales, problems 257-8. Proofs by Catalan, ibid., 1889, 19-22; 1891, 70; 
by G. B. Mathews, Math. Quest. Educ. Times, 52, 1890, 63; by H. J. Woodall, 57, 
1892, 91. 
"Th^orie des nombres, 1891, 420. "Archiv Math. Phys., (2), 11, 1892, 81-3. 
