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

58 F. Luca, L. Szalay 3. Examples Example 1. The equation is a particular case of equation (3) for A = 1, B = — 1 and C — —2. Since B = A + C, all solutions of equation (9) satisfy (Ar + 2) - 2

(n - Ar) = 0, which is equivalent to 2n — 3

A" = 2. The integer solutions of the above equation are given by n — 1 + 'it and Ar = 21 with some integer t, and since n and k must be positive, we must have t > 1. Conversely, one verifies easily that 3í + l\ _ /3í+l\ _

9/3í+1\ 21 ) \2t + 1J \21 + 2J holds for all positive integers t. Example 2. The equation has A = C = 1 and B = 2, therefore D — 0. Moreover, To = Co — 1, so all solutions (n,k) of the above diophantine equation (10) have K + 2

=

t±±H AN

D „_

T = ÍÍ^Ü, which gives t

2 + t - 4 , 2 „ A- = and n = t — 2. Since n > A* > 0, it follows that either t > 3, or i < —3. Conversely, one may check that if t is any integer which is < —3, or > 3, then t

2 _ 2 \ ft

2- 2 (

2 + t —

4 J ~ ~ J + ^ t

2+

t I —