Az Egri Ho Si Minh Tanárképző Főiskola Tud. Közleményei. 1982. (Acta Academiae Paedagogicae Agriensis : Nova series ; Tom. 16)

II. TANULMÁNYOK A TERMÉSZETTUDOMÁNYOK KÖRÉBÖL - Dr. Kiss Péter: Közös elemek másodrendű rekurzív sorozatban

COMMON TERMS IN SECOND ORDER RECURRENCES by Péter Kiss (Summary) Let G = G (A, B, G

0, G[) be a second order linear recurrence defined by rational integers A, B, G

0, Gt and by recursion G

n = AG

n_|— BG

n_2forn > 1. We denote the roots of polynomial x

2 — Ax + B by a and ft. Let us suppose that AB + 0, G

n and G| are not both zero, D = A

2 — 4B > 0, a/ft is not a root of unity and |a[ > |ft\. Let H = H (A, B, Ho. H]) be also a second order linear recurrence with similar conditions. The sequences G and H are called equivalent if there exist integers r and s such that G

n+

r = H

n+

S for every integer n = 0. The following three theorems are proved in the paper. Theorem 1. If the sequences G and H are not equivalent and \ft\ < 1 then the equation G

x = H

y has no solutions with condition x,y>n

{

)-\-l. Theorem 2. If the sequences G and H are not equivalent, B < 0 and \ft\ < 1 then the equation G

x — H

y has no solutions with condition |G

X| =6 mw |B|

n<> Theorem 3. Let the explicit form of the terms of sequences G and H be G

n = (a a

n — b ft

n) /(a —ft) and H

n = (pa

n-q ft

n) / (a — ft) respectively. Suppose that the sequences G and H are not equivalent and ap 4= 0. Then the equation G

x = H

y has no solutions for which x, y > n\. The constants n

0, ni, m and w in the theorems are given in the paper and they depend only on the parameters of sequences G and H. 546