Chap. XXIII]
Single Equations of Degree n>4.
691
Let f=pfq, where p, q are relatively prime. Thus Z is a multiple z 1 q of q
and z = z x p, Zi = z 1 qZ 1 , z 2 = z l qZ 2 .
A. Flechsenhaar 137 and E. Schulte discussed l/a-\-l/b = l/c. E. S6s
(p. 113) treated (2). W. Hofmann 138 discussed the integral solutions of
1 + 1 = 1
a h c’ a b b c
G. Lemaire 139 transformed given decompositions 21// of 9/10 into others.
H. Janculescu 140 noted that in l/x-\-l/y = lfz, z will be integral only when
the g.c.d. d of x and y is a multiple of x/d+yfd.
D. Biddle 141 solved each of 1/(a±h) + 1/ (c±a) = l/a.
Miscellaneous single equations of degree n>4.
J. L. Lagrange 142 noted that, if a is a fixed nth root of unity, the product
of two functions of the type_
p=t+ua^lA+xa 2 ^A 2 -\ 1-za n ~ l Va. 71 “ 1
is of like form. Hence if we replace a by the different nth roots of unity
and form the product of the functions so obtained from p, we obtain a
rational function P of t, u, • • -, z, A such that the product of two functions
of type P is a third function of type P. We can find P by eliminating co
between
œ n ~A = 0, i+nw+Æco 2 + • • • -\-zœ n ~ 1 = l]
then P is the term free of l in the éliminant. For example, if n = 2,
P—t 2 —Au 2 . An application is to the solution of
(1) r n -As n = q m .
We seek to express each factor r—asA lln as an wth power p m , where a n = 1,
and p is the above linear function. Then
p m _ rp_|_ jj a _{_Xa 2 yAH VZa n ~ x ^A 71-1 .
Hence r = T, s = — U, X = 0, • • •, Z = 0. Thus (1) is solvable by this
method if X=0, • • - , Z=0 are solvable. Although only n—2 equations
in n variables, they do not always have rational solutions. For details on
the case n = 3, m = 2, and Lagrange’s extension of the method in his addition
IX to Euler’s Algebra where a n = 1 is replaced by any equation of degree n,
see papers 161-6 of Ch. XXI; also Ch. XX.
Lagrange 143 treated the problem to make y=plq an integer when
p = a+hx-\- • • -, q = a 1 +h 1 x-1 are polynomials in x. By eliminating x,
137 Unterrichtsblatter Math., 16, 1910, 41, 41-2.
138 Ibid., 17, 1911, 14r-15.
139 L’intermédiaire des math., 18, 1911, 214-6.
140 Mathesis, (4), 3, 1913, 119-120.
141 Math. Quest. Educat. Times, (2), 25, 1914, 61-3.
142 Mém. Acad. R. Sc. Berlin, 23, année 1767, 1769; Oeuvres, II, 527-532. Exposition by
A. Desboves, Nouv. Ann. Math., (2), 18, 1879, 265-79; applications, 398-410, 433-444,
481-499; also by R. D. Carmichael, Diophantine Analysis, New York, 19Î5, 35-63.
Cf. Dirichlet 19 ; also Libri 64 - 65 of Ch. XXV.
143 Addition IV to Euler’s Algebra, 2, 1774, 527-533. Oeuvres de Lagrange, VII, 95-8.
Euler’s Opera Omnia, (1), I, 579.
