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

Linear diophantine equations with three consecutive binomial coefficients 57 for some positive integer t. (Hi) B / A + C, D = B

2 - 4AC > 0 is not a perfect square, and (5) x

2 - m'

2 = E holds, where X = (B

2 - 4/lC)(/?. - A*) - A(B - 2C), F = + 2) + B(n - fc) - A, E — 4A

2C(A — B + C), case in which all positive integer solutions (n

, k) of equation (3) can be found by solving the Pell like equation (5). Proof. After simplifications, equation (3) becomes A(k 4- l)(Jfc 4- 2) 4- B(k + 2)(n - k) + C(n - k)(n - k - 1) = 0. Writing k + 2 = x, n — k — y we get We shall assume that D B

2 — A AC ^ 0, and we shall return to the case when D — 0 later. With the substitution x = u 4- et, y = v 4- ß

, we get that the above relation becomes We choose a and ß such that the coefficients of the linear terms in u and v in equation (7) vanish. These lead to the system of equations Ax(x -1)4- Bxy 4- Cy(y — 1) = 0, or, equivalently, (6) Ax

2 4- Bxy + Cy

2 - Ax - Cy - Ü. (7) (Au

2 4- Buv 4- Cv

2) 4- (2,4a + Bß - A)u + (Bo 4- 2

Cß - C)v = -(Aa

2 4- Baß 4- Cß

2) 4- Aa 4- Cß. 2 Aa Bß — A Ba 4- 2

Cß = C, whose rational solution is C(B - 2.4) B

2 - 4AC ' ß = A(B — 2 C) B

2 - 4 AC