and a new Theorem concerning Prime Numbers. 131 
lowing, which is itself an immediate corollary from the arith- 
metical theorem that if a,b,c, ... 1, with or without repetitions, 
are the distinct prime factors of the denominator of a fraction, 
the fraction itself may be resolved into the sum of simple frac- 
tions, 
ete By oC L 
ee a OT 
(itself a direct inference from the familiar theorem that if 
P; 7 be any two relative primes, the equation pa—qy=c is 
soluble in integers for all values of c). The lemma in question 
is as follows: If the quantity above described is representable 
under the several forms, 
al ; b i! 
ot an (a) integer yet 8 (6*) inteser-e zeta (A) integer, 
then it is equal to 
ahs: Bl i 
epage th et ad absolute integer. 
From what has been already shown, it is obvious that « being 
any prime number, the highest power of « which can enter into 
the denominator of (u?”—1)B, is y?”, and consequently y?"B is 
an integer relative to uw. Also it is clear that only those values 
of ~ can appear in the denominator of B, which, diminished by 
unity, are factors of 2n. We have, moreover, 
1 
(—)"*(w"—1)B, =I (2n) x coefficient of #2”! in wt ea 
2. e. coefficient of ¢2"-! in eri Ae where 
N =I (2n)(e™—Dé4 eu—2#4 ... +e’—(u—1)) 
=V,t +57 + eee + Vol?” + &e., 
as is quite proper, for both of them are fractions in their simplest forms, 
which would not be the case for the former were ¢ equal to or greater than p, 
since in such case S could be more simply expressed under the form 
me sy. 
This principle amounts to an affirmation that the equation in positive 
integers, 
(0...kl)e+(ab...2y+...+(ab...k)t—(ab...klu=N, 
where a, b,..k,/ are relative primes, and N<(ad... hi), always admits of a 
solution, which may be termed the primitive one, and which will be unique, 
that namely in which 2, y,...2, ¢ are respectively less than a, b,..k, 1. 
