Az Eszterházy Károly Tanárképző Főiskola Tudományos Közleményei. 1997. Sectio Mathematicae. (Acta Academiae Paedagogicae Agriensis : Nova series ; Tom. 24)

LIPTAI, K. and TÓMÁCS, T., Pure powers in recurrence sequences

38 By (1) we have Kálmán Liptai and Tibor Tómács (1 + e

n) (1 + e

x) n (1 + ^

) a

k+

2a

n+x+

x>

++

x" = w

* i=i from which (3) qlog w = (k + 2)log a + ^n + x + ^ log a + log (1 + £

n) k + log(l + £

x) + ^bg(l + e

x.) i=

1 follows. It is obvious that x < n-\-x(- ^ X{ < (k + 2)x

. Using that log |1 -f£

7 i=i is bounded and lim -ri(m)(

SL

i-)

m = 0 (i = 2,..., 5), we have m —i rv^ ^ \ Ot / (4) X X CL — < log w < c

2 q q where C\ and c

2 are constants. Let L be defined by L := log G

nG

X

l G

x

2 • • • G

X

k aa

3 = |log(l + £

X)\. By the definition of £

x and the properties of logarithm function there exists a constant C3 that (5) L < e -c

3x On the other hand, by the Lemma with v = k + 4, Mk+

4 = w,B' = q and B — x we obtain the estimation (6) L= q log to —log log G

X

i -log a-x log a ^

>

e-C(lo

g q log uj+x/q) where C depends on heights. By x^ < Kn heights depend on G

n,

..., Gj<

n> i.e. on n, Ii,k and on the parameters of the recurrence. By (4), (5) and (6) we have c^x < C(log <7 log w + x/q) < C4 log q log w

, i.e. (7) x < C5 log q log w