272 HlSTOKY OF THE THEORY OF NUMBERS. [CHAP. IX J. Wolstenholme78 noted that the highest power of 2 dividing (2m^ *) is q—p — 1, where q is the sum of the digits of 2m —1 to base 2, and 2P is the highest power of 2 dividing ra. J. Petersen79 proved by Legendre's formula that (a+b) 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. Cesaro80 treated Kummer's71 problem. He stated (Ex. 295) and Van den Broeck81 proved that the exponent of the highest power of the prime p dividing (2nft) is the number of odd integers among [2n/p], [2n/p2], /p*],.... O. Schlomilch81a stated in effect that (n+i) is divisible by n. E. Catalan82 proved that if n is odd, W. J. C. Sharp820 noted that (p+ri)l-p\nl is divisible by p2, if p is a prune >n. This follows also from (ptn)=l (mod p) [Dickson90]. L. Gegenbauer83 noted that, if <r is any integer, r one of the form 6s or 3s according as n is odd or even, n-l) =0 (mod n+2)" The case n odd, cr=2, r = 3, gives Catalan's result. E. Catalan84 proved Hermite's75 theorem. Ch. Hermite85 stated that (£) is divisible by m—n+1 if m is divisible by n; by (m— n-f l)/e if € is the g. c. d. of w-f-1 and n\ by ra/5, if d is the g. c. d. of m, n. E. Lucas86 noted that, if n^p — 1, p— 2, p — 3, respectively, if p is a prime, and proved Hermite's75 result (p. 506). F. Rogel87 proved Hermite's75 theorem by use of Fermat's. 78Jour. de math. 616m. et sp6c., 1877-81, ex. 360. 78Tidsskrift for Math., (4), 6, 1882, 138-143. 8°Mathesis, 4, 1884, 109-110. 81im, 6, 1886, 179. 81fflZeitschrift Math. Naturw. Unterricht, 17, 1886, 281. 82M<§m. Soc. Roy. Sc. de Liege, (2), 13, 1886, 237-241 ( = M<§langes Math.). Mathesis, 10, 1890, 257-8. 82«Math. Quest. Educ. Times, 49, 1888, 74. 83Sitzungsber. Ak. Wiss. Wien (Math.), 98, 1889, Ha, 672. ""MSm. Soc. Sc. Li&ge, (2), 15, 1888, 253-4 (Melanges Math. III). 86Jour. de math. spe"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. 86The"orie des nombres, 1891, 420. 87Archiv Math. Phys., (2), 11, 1892, 81-3.