CHAPTER XL MISCELLANEOUS THEOREMS ON DIVISIBILITY, GREATEST COMMON DIVISOR, LEAST COMMON MULTIPLE. THEOBBMS ON DIVISIBILITY. An anonymous author1 noted that for n a prime the sum of 1, 2,..., n—1 taken by twos (as 1+2, 1+3,.. .)> by fours, by sixes, etc., when divided by n give equally often the residues 1, 2,..., n-1, and once oftener the residue 0. The sum by threes, fives,..., give equally often the residues 1,..., n—1 and once fewer the residue 0. J. Dienger2 noted that if m2^1*! andCm^2-!)/^2-!) are divisible by the prune p, then the sum of any 2r+l consecutive terms of the set 1, w2*, w2'2", w3*2",... is divisible by p. The case w=2, r = l, p = 3 or 7 was noted by Stifel (Arith. Integra). G. L. Dirichlet3 proved that when n is divided by 1, 2,..., n in turn the number of cases in which the remainder is less than half the divisor bears to n a ratio which, as n increases, has the limit 2—log 4 = 0.6137 ...; the sum of the quotients of the n remainders by the corresponding divisors bears to n a ratio with the limit 0.423.... Dirichlet4 generalized his preceding result. The number h of those divisors 1, 2,..., p (p ^ n), which yield a remainder whose ratio to the divisor is less than a given proper fraction a, is Assuming that p2/n increases indefinitely with n, the limit, of h/p is a if n/p increases indefinitely with n, but if n/p remains finite is n ]f rn "|1 * J. J. Sylvester6 noted that 2m+l is a factor of the integral part of and of the integer just exceeding &2m, where fc = l + \/3. N. V. Bougaief6 called a number primitive if divisible by no square >1, secondary if divisible by no cube. The number of primitive numbers ^ n is where q(u) is zero if u is not primitive, but is +1 or —1 for a primitive u, according as u is a product of an even or odd number of prime factors. your, fiir Math., 6, 1830, 100-4. 2Archiv Math. Phys., 12, 1849, 425-9. 3Abh. Ak. Wiss. Berlin, 1849, 75-6; Werke, 2, 57-58. Cf. Sylvester, Amer. Jour. Math., 5, 1882, 298-303; Coll. Math. Papers, IV, 49-54. 4Jour. fur. Math., 47, 1854, 151-4. Berlin Berichte, 1851, 20-25; Werke, 2, 97-104; French transl. by O. Terquem, Nouv. Ann. Math., 13, 1854, 396. "Quar. Journ. Math., 1, 1857, 185. Lady's and Gentleman's Diary, London, 1857, 60-1. "Comptes Rendus Paris, 74, 1872, 449-450. Bull. Sc. Math. Astr., 10, I, 1876, 24. Math. Sbornik (Math. Soc. Moscow), 6, 1872-3, I, 317-9, 323-331.