Az Egri Ho Si Minh Tanárképző Főiskola Tud. Közleményei. 1987. (Acta Academiae Paedagogicae Agriensis : Nova series ; Tom. 18/11)

Kiss Péter: A Lucas számok prímosztóiról

- 17 KISS PÉTER A LUCAS SZÁMOK PRÍMOSZTÓIRÓL* Abstract: (On prime divisors of Lucas numbers) Let R be a non-degenerate Lucas seguence defined by R

n

=

=A. R

n_

l+B. R

n_

2 <n > 1) , where R

o=0, R

t=l and A, B are fixed integers. Donate by r(p) the rank of the apparition of a prime p in the sequence, that is p I R but p \ R for 0 < m < rCp) , It is known r (p) m that if p is a prime with (p,B)=l, then r(p) exists, (i.e. there are terms divisible by p) and rCp) | jp-CD/piJ , where (D/p) is the Legendre-symbol and D=A

2+4B . In an earlier paper we proved that [p-CD/p)j^/rCp) is undbounded if p tends to infinity. Now we show: (i) for almost all primes p we have [p-CD/p)j^/rCp) > q[r(p)J , where q(x) is a non decreasing arithmetical function with qCrO/log n —> 0 if n —* <» ; (ii) for any 6 with 0 < 6 < 1 the set of primes p for which [p-CD/p)J y/rCp) > Ó .log p [or rCp) < p/C<5. log p)J has positive density in the set of all primes; (iii) the set of integers n, for which each primitive prime divisor of R

n is is greater than ó.n.log n (where 0 < ó < 1 ), has positive density (p is a primitive prime divisor of R_ if r(p)=n). * A kutatást (részben) az Országos Tudományos Kutatási Alap 273. sz. pályázata támogatta.