C OMPOSITIO M ATHEMATICA
R. T IJDEMAN
On the maximal distance between integers
composed of small primes
Compositio Mathematica, tome 28, no 2 (1974), p. 159-162
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159
COMPOSITIO MATHEMATICA, Vol. 28, Fasc. 2, 1974, pag. 159-162
Noordhoff International Publishing
Printed in the Netherlands
ON THE MAXIMAL DISTANCE BETWEEN INTEGERS
COMPOSED OF SMALL PRIMES
R.
Tijdeman
1 n2 ··· be the
Let p1, ···, pr be fixed primes, r ~ 2. Let n 1
sequence of all positive integers composed of these primes. In [3] we
proved the existence of an effectively computable constant Cl &#x3E; 0 such
that
=
In this note
stants
C2
&#x3E;
shall prove the existence of
0 and N such that
we
effectively computable
The average order of the difference ni+ 1-ni is about
con-
ni/(log ni)r-1.
Hence, C1 ~ r -1, C2 ~ r -1. (Compare [3].)
In the
fractions
of (1) we use some elementary properties of continued
for
(See example [2, Ch. V]) and a result of N.I. Fel’dman. All
constants c, cl , C 2, ... will be positive and effectively computable. They
only depend on the fixed primes p1, ···, pr or p, q.
proof
1
LEMMA:
Let p, q be fixed primes, p :0
of convergents of log pllog q.
computable constant c such that
sequence
PROOF: One
Since
we have
has kj ~
2
for j ~ 2.
q. Let ho/ko, hl ikl,
Then there exists an
... be the
effectively
160
On the other
hand, Fel’dman’s result [ 1] implies
where Ho = max (1 + hj, 1 + kj) and Cl is a constant. Since hj/kj is
bounded by 1 + log p/log q, one has 1 + log H0 ~ c2 log kj for j ~ 2.
So we obtain a constant c such that
The lemma follows from
(2)
and
(3).
2
In order to prove (1) we may
Hence, it suffices to prove
assume r
=
2 without loss of generality.
THEOREM: Let p and q be primes, p ~ q. Let n1 = 1
n2 ... be the
these
all
positive integers composed of
primes. Then there
sequence of
exist effectively computable constants C and N such that
PROOF: Let n = ni
pu ~
~n,
=
puqv ~
N. It is
no
restriction to
assume
and, hence,
Let ho/ko, h1/k1, ... be the convergents of log p/log q. Then
is a monotonic increasing sequence. Take j such that kj ~ u
suppose that N is so large that both n ~ 3 and j ~ 2. We
cases
(a)
Put n’ =
that
and
(b).
pu-kjqv+hj. Hence, n’
&#x3E; n.
It follows that
Using (4) and u
kj+1
we
obtain
We have
kl , k2, ···
kj+1. We
distinguish
161
We see that n’/n has an upper bound only depending on p and
fore have
for
some
constant c3.
q. We there-
The combination of these inequalities yields
and, hence,
Then hj-1/kj-1
&#x3E;
log p/log
q. Put n’
=
pu-kj-1qv+hj-1.Hence,
We have
It follows that
We know from the lemma that
Using (4) and
u
kj, 1
we
obtain
Hence,
for
some constant
cs. It follows that
n’
&#x3E; n.
162
We have in both cases, from
For N
This
sufficiently large
this
(5) and (6),
implies
completes the proof.
REFERENCES
[1] N. I. FEL’DMAN: Improved estimate for a linear form of the logarithms of algebraic
numbers. Mat. Sb. (N.S.) 77 (119) (1968) 423-436. Transl. Math. USSR Sbornik
6 (1968) 393-406.
[2] I. NIVEN: Irrational numbers. Carus Math. Monographs, 11. Math. Assoc. America.
Wiley, New York, 1956.
[3]R. TIJDEMAN: On integers with many small prime factors. Compositio Mathematica
26 (1973) 319-330.
(Oblatum 5-X-1973)
Mathematical Institute
University of Leiden
Leiden, Netherlands.