Az Eszterházy Károly Tanárképző Főiskola Tudományos Közleményei. 2004. Sectio Mathematicae. (Acta Academiae Paedagogicae Agriensis : Nova series ; Tom. 31)

LUCA, F. and SZALAY, L., Linear diophantine equations with three consecutive binomial coefficients

Acta Academiae Paedagogicae Agriensis, Sectio Mathematicae 31 (2004) 53-60 LINEAR DIOPHANTINE EQUATION WITH THREE CONSECUTIVE BINOMIAL COEFFICIENTS Florian Luca* (UNAM, Mexico) László Szalay (Sopron, Hungary) Abstract. In this note, we study the diophantine equation + ; + I)+C(*

+2)—

0 positive integers (n,fc), where A, B and C are fixed integers. AMS Classification Number: 11D04, 111)1)9 1. Introduction D. Singmaster (see [3]) found infinitely many positive integer solutions (n,k) to the diophantine equation All such solutions arise in a natural way from the sequence of Fibonacci numbers {F

m)

m>

0 given by F

0 = ü, F\ = I and F

m+

2 = F

m+

i + F

m for rn > 0. Goetgheluck (see [1]) extended the above result and found infinitely many positive integer solutions (n,k) for the diophantine equation These solutions arise in a natural way from the positive integer solutions of the Pell equation x

2 — 3

y

2 — —2. Several other diophantine equations involving binomial coefficients have been considered in [2], [4] and [5]. In this note, we fix three integers A, B, C, not all zero, and look at the positive integer solutions (n,k

) of the equation + B(

k'^_

[) + C(

k,?

2) = 0. To avoid degenerate cases, we shall assume that l<k<.k + 2<n — 1. We shall also assume that AC 0. Indeed, say if 4 = (J. then the above equation simplifies to * This research was partially sponsored by grants SEP-CONACYT 37'259-E and 37260-E.