An introduction to the modern theory of equations, by Florian Cajori.

BINOMIAL AND RECIPROCAL EQUATIONS

77

be shown that if un-1 and Yu are any two successive conver.
Vn —1 Vl
gents, then u'nv,_l - u,,_~V11n = ~ 1.) But
m ==p(Cr + 1) + r, n = r + 1;
hence, if we take a =pq - 1, b = q, we have
mb - CacI = i 1. Q.E.D.
Now, if possible, let a be a root common to x" - 1 = 0 and
x1 - 1 =0. Then (0n = 1, n = 1 and amb = 7l,1 (. = 1, where a
and b are numbers which satisfy the relation mb - na= 1.
Hence, ecmb-" = 1, a-=l1, or a =1. That is, 1 is the only
root common to the two equations.
IV. If h is the highest common factor of m and n, then roots
of x' 1 - = 0 are common roots of x"m -- 1 = 0 and x" - 1 = 0.
We have m = hm', n = hin', where n' and n' are prime to each
other. Hence it is possible to find integers a and b. such that
m'b - n'a = 1. Multiplying by h, we get mb - na = - h.
Now, if a is a common root, we have c 1, = 1., -'na =, b- = 1,
or a-:' = 1. This means that a is a root of h - 1 = 0.
V. If a is a complex root of 1x" - 1 = 0, t beirg prime, then
the roots are 1, ca, a2,,..^, * n -~
By II, 1, a1,..., a'-,r/ are all roots of the equation. They
are all different; for suppose. = r-, then,e - =. But by
III, -- 1 = 0 and X- - -1 = 0 cannot have a root in conmmon,
since n and ( — q) are prime to each other. Hence the equation a(P- = 1 is impossible, and all the roots are included in
the series 1, a,..., w'.-l
VI. The roots of the equations
P-1=0, x q-1 =0, ) - 1 = 0,..
all satisfy the equation xpr... - 1 = 0,
For if a is a root of P - 1 = 0, then at = 1 andc (Cp)q"'" = 1,
or c'q'"' = 1. That is, az is a root of xpqi' -_ 1 = 0.