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 Example 3. The equation reduces to equation (3) for .4=1, B — 1, and C — —1. In this case, I) = B

2 - 4AC = 5, = 4A

2C(A — B + C) = 4, X = (B

2 -4AC){n —k) - A(B-2C) = 5(n - k) - 3, and Y = + 2) + B{n - k) - A = 2(k + 2) + (« - k) - 1. Since X

2 — 5

Y

2 = 4, it follows that A' = L

m and Y = F

m hold with some even positive integer m, where (Le)e>o is the Lucas sequence given by LQ = 2, L\ = 1, and Li+

2 = ^i+i + Li for all t > 0, and (Fc)oo is the Fibonacci sequence. We now get that n - k = (X + 3)/5 = (L

m + 3)/5, and that k + 2 = (F - (n - + l)/2 = (5F

m - X

m +2)/10. Hence, A; = (5F

m - L

m - 18)/10, and n = (5F

m+L

m - 12)/10. Since Ii and A- are integers, we need that 5|L

m + 3, and that 10|5F

m — L

m + 2. Thus, 5|L

m + 3 and 2|F

m -f L

m. The second relation is always fulfilled, while the first one is fulfilled precisely if m = 0 (mod 4). Thus, n = {bF^ + L\t — 12)/10, and k = (5 F 41 — L4t — 18)/10. Since k > 0, we also need that 5F.i

f > L

+ 18, which forces t > 2. One can now easily verify that 5F.,, + /..„-1

2x / 5 f4 > + /- •» f — 12 10 \ / 10 \ , / 10 -l-u+

2 I I 5/.\,,-L

4,8 I ^ I 5t — L.u— 18 10 / v 10 / x 10 holds for all integers t > 2. Note also (hat since n \ {n\ (n + 1 k+ [J \kj \k+ 1 it follows that the diophantine equation (11) reduces to the diophantine equation (11), which in turn is a consequence of our Theorem. Remark. We remark that at instance (iii) of our Theorem, it could be possible that the Pell equation (5) has integer solutions (A", F), and yet none such that the additional congruence X = — A(B — 2C) (mod B

2 — 4AC) (necessary in order for n — k to be an integer) is satisfied. References [1] GOETGHELUCK, P., Infinite families of solutions of the equation (£) = 2(£), Math. Comp. 67 (1998), 1727-1733. [2] LUCA

, F. Consecutive binomial coefficients in Pythagorian triples, The Fibonacci Quart. 40 No. 2 (2002), 76-78. [3] SiNGMASTER, D., Repeated binomial coefficients and Fibonacci numbers, The Fibonaci Quart. 13 (1975), 295-298.