30 Stewart, On divisors of linear recurrence sequences

Certainly the left hand side of inequality (62) is at least II——jlog« for n ciently large and thus, since we may assume that 0<б<1, m is at least C23]/logn, Since m = t-\'y and y<C24 we deduce that

( 63 ) M--^jlog«<n+^m

for n sufficiently large. By the arithmetic-geometric mean inequality

П logp.^l

1=1

Since TiPi^Qitiin)), it follows from (63) that

i=i

- ^j log« <^ tlogt + t log logo («(«)).

If we assume that / is less than il—-z-]z—;----- then tlogt is less than log« hence

\ 5/loglog«

1 —r- jlog« <r log logQ{u{n)),

from which it follows that Ô(w(«))>«, as required. Thus we may assume that t is at

least I 1-----J-----— and in this case the product of the first t primes is at least

\ 5/loglog«

^i - e jPqj. ^ sufficiently large. Therefore,

Q { u { n ) ) ^nPi>n' - \

i=l

and this establishes (9).

For the proof of (8) we may assume that p^ is less than log«. As a consequence the right-hand side of (63) is less than (l+—jHogr, whence (1—-|log«<r logr,

for n sufficiently large. Thus

/ g \ log« \ 3/log log«*

Certainly p^ is greater than or equal to the Mb prime number and so by the prime number theorem

^> ( l - e ) log« , for n sufficiently large. Since Р(и{п})^р^, this completes the proof of the theorem.