332 History of the Theory of Numbers. [Chap, xi 
between the QAz), relations involving the number A{z) of primes ^z, and 
relations involving both Q's and A's. 
A. Rothe'*^^ called b a maximal divisor of a if no larger divisor of a con- 
tains 6 as a factor. Then a/b is called the index of b with respect to a. 1| 
If also c is a maximal divisor of b, etc., a,b, c, . . ., I 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]. 
*Weitbrecht-^ noted tricks on the divisibility of numbers. 
*E. Moschietti-^ discussed the product of the divisors of a number. 
Each-^ 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. Verson^^ noted that the consecutive numbers 1375, 
1376, 1377 are divisible by 5^ 2\ 3^ respectively. 
J. G. van derCorput^^ evaluated 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^^ et seq. in Ch. XVII; also Binet^^ and Dupre^. 
V.A.Lebesgue^^notedthatthel.c.m.of a, . . .,A;is(p]P3P5. . ■)/{v2ViP&- ■ ■) 
if pi is the product of a, ... , k, while p2 is the product of their g. c. d.'s two 
at a time, and ps the product of their g. c. d.'s three at a time, etc. If a, 6, c 
have no common divisor, there exist an infinitude of numbers ax-^b rela- 
tively prime to c. 
V. Bouniakowsky^^ determined the g. c. d. N of all integers represented 
by a polynomial /(x) with integral coefficients without a common factor. 
Since A^ divides the constant term of f{x), it remains to find the highest 
power p" of a prime p which divides J{x) identically, i. e., for x = 1, 2, . . . , p". 
Divide /(x) by Xp={x — 1). . .{x — p) and call the quotient Q and remain- 
der R. Then must R^O (mod p") for x = l,. . ., p, so that each coefficient 
of R is divisible by p", and iu = Mu vvhere p"' is the highest power of p divid- 
ing the coefficients of i?. If /ii = l, wehaveju= 1. Next, let /ii>l. Divide 
^^Zeitschrift Math.-Xaturw. Unterricht, 44, 1913, 317-320. 
"Vom Zahlenkunststiick zur Zahlentheorie, Korrcspondenz-Blatt d. Schulen Wiirttembergs, 
Stuttgart, 20, 1913, 200-6. 
"Suppl. al Periodico di Mat., 17, 1914, 115-6. 
i^Math. Quest. Educ. Times, 36, 1881, 48. 
2'Math. Miscellany, Flushing, N. Y., 1, 1836, 370-1. 
"Nieuw Archief voor Wi.skunde, (2), 12, 1918, 213-27. 
"Jour, de Math., (1), 6, 1841, 453. 
»Ibid., (1), 11, 1846, 41. 
'*Nouv. Ann. Math., 8, 1849, 350; Introduction k la th6orie des nombres, 1862, 51-53; Exercises 
d'analyse num^rique, 1859, 31-32, 118-9. 
"M<5m. acad. sc. St. P^tersbourg, (6), ac. math, et phys. 6 (so. math. phys. et nat. 8), 1857 
305-329 (read 1854); extract in Bulletin. 13, 149. 
