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 59 It is clear that the first one admits no integer solutions x = k + 2 and y = n — k for 1 <A*<Ar + 2<n — 1, while the second one admits infinitely many such solutions if and only if C < 0 (whereas if C > 0. then the second one does not admit any such solutions either). Finally, if E ^ 0, then equation (8) admits only finitely many solutions (or none) if D is a perfect square, while if D is not a perfect square, the above equation (8) is a Pell like equation, which either has no solutions, or it has infinitely many, and in this later case all integer solutions (A', Y) of such equation belong to finitely many binary recurrent sequences whose roots are the fundamental unit ( of norm 1 in the quadratic order IK = <Q[s/D] a. id its conjugate Ci, respectively. Finally, we deal with the case I) = 0. In this case, B

2 — 4AC, so B - '2B

0

, and Bl - AC. Since gcd(.4, B, C) = 1, and A > 0, it follows that gcd(/l,C) = 1. and then that A = AQ and C = CQ hold with some positive integers Ao and Co- Hence, BO = ±AQCQ. When BO — AQCQ

, it is clear that the left hand side of equation (3) is positive whenever 1 < k < k + 2 < n — 1. Thus, Bo — — -4oC'o, and therefore B = —IAqCo- Equation (6) becomes A

2

0X

2 - 'lAoCoxy + Cly

2 = A

2

0x + C^y, which can be rewritten as {A

0x - C'oy)

2 = A

2x + Cly = A

0(A

0x - C

0y) + C

0(A

0 + C

0)y. Setting t Aox — Coy

, we get that Co(Ao + C

0)y = t

2 - A

0t, leading to = t(t-Ap) y Co(Ao + Co)

' and since Aox = Coy + t, we get that t

(t + Cp) Ao{Ao + C

oy which lead to formulae (4) via the fact that x = k + 2, and y = n — k. Note that since x, t, and y are integers, it follows that t is in certain arithmetical progressions modulo AoCo{Ao + Co), and from the fact that x > 3 and y > 3, it follows that either t > G\ := Gi(Ao,Co), or t < Gi := GI{AQ, CO), where G\ and G

2 are two constants which depend on ,4o and Co and which can be easily computed by solving the coresponding quadratic inequalities. This completes the proof of the Theorem.