HISTORY OP THE THEOKY or NUMBEKS. [CHAP, in
which reduces by cancellation to (1 + Pxz)(l + P*lz) •••(!+
if n is a positive integer. The simplest of the general formulas proved is
which includes as special cases formulas of Euler3* 9 and Cauchy.29
A. S. Werebrusow188 gave a recursion formula for the number of sets of positive integral solutions of aiXi + • • • + o,nxn = A, where the positive integers a have no common factor. Then he considered the number of sets when at least one x is ^ 0.
P. A. MacMahon189 treated a " general magic square," consisting of n2 integers (zeros and repetitions permitted) arranged in a square such that the rows, columns and diagonals contain partitions of the same number (whereas' in an ordinary magic square the n2 integers are 1, 2, • • •, n2). The treatment applies to all arrangements of integers which are defined by linear homogeneous Diophantine equations or inequalities such that the sums of corresponding elements of two solutions give a solution [cf . MacMahon174].
0. Meissner190 noted that to decompose n into positive integral sum-mands whose product is a maximum, the summands must be equal or differ at most by unity, and must include as many threes as possible.
G. Mignosi191 wrote c» for the number of sets of integral solutions ^ 0 of 0,1X1 + • • • + amXm = ft, and <r(j) for the sum of those of ai, • • •, am which are divisors of j, and proved the recursion formula
<r(l)Ci-i + <r(2)C;_2 + • • • + or(i)cb = iCi, Co = 1.
Taking i — 1, • • •, ft, we obtain nlcn as a determinant of order n. If eacl| a» = 1, then <r(i) = m and cn is the number of combinations of m + n — 1 things taken n at a time.
S. Minetola192 wrote Rm> n for the number of different ways m distinct objects can be separated into n groups, where n ^ m. For example, #4,2 = 7, the separations being ai — a2a3a4, • • •, a* — a^as, ctia2 — ai£z3 — #004, 01^4 — #203. We have
Rm, n = nRm-lt n + Rm-l, n-1, Rm, 2 = 1 + 2 + 22 + • • - +
^-^
\m — k —
k, k,
88 Matem. Sbornik (Math. Soc. Moscow), 24, 1904, 662-688.
18 Phil. Trans. Roy. Soc. London, 205, A, 1906,37-59. Memoir III on Partitions. Abstract in Proc. Roy. Soc., 74, 1905, 318.
90 Math. Naturw. Blatter, 4,1907, 85.
91 Periodico di Mat., 23,1908,173-«.
92 Giornale di Mat., 45,1907, 333-366; 47,1909,173-200. Corrections, generalizations and
simplifications in II Boll, di Matematica Gior. Sc.-Didat., Rome, 11, 1912, 34-50, with errata corrected pp. 121-2.