Az Eszterházy Károly Tanárképző Főiskola Tudományos Közleményei. 1989. 19/8. (Acta Academiae Paedagogicae Agriensis : Nova series ; Tom. 19)

Kiss Péter — Bui Minh Phong: Reciprocal sum of prime divisors of Lucas numbers

MÁTYÁS FERENC PITAGORASZI SZÁMHÁRMASOK fö A LUCAS SOROZAT k AUSTRAG

T CPyi hagorean triples and ihe l.ucr irt sequenced Ve define ihe sequence L=mL ^ bv ihe iníegcrB L--2, L =1 and ihe recurrence L »L. _ L _. n >1. O'i ri n-i rí- 2 This sequence is called Lucas sequence and the terms of il are ihe Lucas number ft. x o

' ^ o ' ^ o positive integers are called Pythagorean triple if = z

2. If for this J o J o o 3 o

' ^ o o triple (

x

0»y

0»

z

0]=l ai«« t™«? then x

0>y

0'

Z0 triple will be called primitive Pythagorean triple. In this paper we deal with ihe connection between the Lucas numbers and the Pythagorean triples. Let A

sn,C CAs^O) be arbitrary, but fixed integers. Ve prove ihe following two theorems: THEOREM 1

, CAL ,L + B, L >C> triples are ——— — r> 2 n Zn ' Pythagorean iriples for every even n(?03 if and only if Á = 0 Cmod 2), B = - j +2 and C - j jj

2+2 , v J j - y y while for every odd nC^l) t/ and only if A s 0 Cmod 2), B - -f^J

2-2 and G » (Éj

2"

2 ' THEOREM

JZ. Under ihe conditions of Theorem 1. ihe AL^ , L +B

P L

2, +G I iriples are primil ive Pythagorean