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

54 F. Luca, L. Szalay Obviously, equation (2) has no solution if BC > 0. Suppose that BC < 0 (say, up to changing signs, that B < 0 and C > 0) and that gcd{B,C) = 1. Then equation (2) implies B(k + 2) -f- C(n — k — 1) = 0, which can be rewritten as n = ((C - B)k + C - 2B)/C = k + 1 - B(k + 2)/C. Thus, n is an integer if and only if A: = —2 (mod C). Moreover, the conditions l<Ar<Ar + 2<n — 1 are always fulfilled if k > 1 and k > —2(1 + C/B), and therefore (2) has infinitely many solutions. The case when (7=0 can be reduced to the case when A = 0 by using the symmetry of the binomial coefficients and the substitution (A

, C, k) i

—> (C

, n — k - 2). Acknowledgements. This paper was written during a very enjoyable visit by the first author to University of West Hungary in Sopron; he wishes to express his thanks to that institution for the hospitality and support. 2. Main Result It is clear that we may assume that gcd(A,B,C) = 1 and that A > 0. Our main result is the following. Theorem. Let A

, B and C be integers with A > 0, C ^ 0 and gcd(.4, B

, C

) = 1. If the diophantine equation <

3> ;,)=•• admits infinitely many integer solutions l<k<k + 2<n — 1, then one of the following holds: (i) B = A + C and C < 0, case in which all the solutions (n, k) are on the line A(k + 2) + C{n - k) = 0, (ii

) A = AQ

, B = — 2_4QCo, C = C'% hold with some positive coprime integers AQ and Co, case in which all solutions (n

, k) with l<k<k + 2<n — 1 of (3) are of the form (4) k + 2 = ; —— and n - k = Ao(Ao + Co) Co(Ao + Co)