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332 HISTORY OF THE THEORY OF NUMBERS. [CHAP.XI between the Qx(z), relations involving the number A(z) of primes ^z, and relations involving both Q's and A's. A. Rothe23 called b a maximal divisor of a if no larger divisor of a contains 6 as a factor. Then a/6 is called the index of b with respect to a. If also c is a maximal divisor of 6, etc., a, 6, c,..., 1 are said to form a series of composition of a. In all series of composition of a, the sets of indices are the same apart from order [a corollary of Jordan's theorem on finite groups applied to the case of a cyclic group of order a]. *Weitbrecht24 noted tricks on the divisibility of numbers. *E. Moschietti25 discussed the product of the divisors of a number. Each26 of the consecutive numbers 242, 243, 244,245 has a square factor > 1; likewise for the sets of three consecutive numbers beginning with 48 or 98 or 124. C. Avery and N. Verson27 noted that the consecutive numbers 1375, 1376, 1377 'are divisible by 53, 23, 33, respectively. J. G. van derCorput28evaluated the sum of thenth powersof all integers, not divisible by a square >1, which are ^x and are formed of r prime factors of m. GREATEST COMMON DIVISOR, LEAST COMMON MULTIPLE. On the number of divisions in finding the g. c. d. of two integers, see Lame"11 et seq. in Ch. XVII; also Binet33 and DuprS34. V.A.Lebesgue35notedthatthel.c.m.of a,..., k is (pipzps. • O/CPzWe- • -) if pi is the product of a,.. ., k, while p2 is the product of their g. c. d.'s two at a time, and p3 the product of their g. c. d.'s three at a time, etc. If a, b, c have no common divisor, there exist an infinitude of numbers ax+b relatively prime to c. V. Bouniakowsky36 determined the g. c. d. N of all integers represented by a polynomial f(x) with integral coefficients without a common factor. Since N divides the constant term of /(re), it remains to find the highest power p" of a prime p which divides / (x) identically, i. e.} for x = 1, 2,..., pM. Divide f(x) by Xp= (05-1)... (x—p) and call the quotient Q and remainder R. Then must #s=0 (mod pM) for x= 1,..., p} so that each coefficient of R is divisible by p", and ju^/^, where p*1 is the highest power of p dividing the coefficients of R. If ju: = 1, we have JJL = 1. Next, let /zx > 1. Divide 23Zeitschrift Math.-Naturw. Unterricht, 44, 1913, 317-320. 24Vom Zahlenkunststttck zur Zahlentheorie, Korrespondenz-Blatt d. Schulen Wiirttembergs, Stuttgart, 20, 1913, 200-6. "Suppl. al Periodico di Mat., 17, 1914, 115-6. 26Math. Quest. Educ. Times, 36, 1881, 48. 27Math. Miscellany, Flushing, N. Y., 1, 1836, 370-1. 28Nieuw Archief voor Wiskunde, (2), 12, 1918, 213-27. 33Jour. de Math., (1), 6, 1841, 453. «/Wd., (1), 11, 1846, 41. 36Nouv. Ann. Math., 8, 1849, 350; Introduction a la the"orie des nombres, 1862, 51-53; Exercises d'analyse numerique, 1859, 31-32, 118-9. m. acad. sc. St. Petersbourg, (6), so. math, et phys. 6 (sc. math. phys. et nat. 8), 1857 305-329 (read 1854); extract in Bulletin, 13, 149.