László Szalay

Generalized balancing numbers

Abstract The positive integer x is a (k, l) -balancing number for y(x ≤ y — 2) if 1k + 2k + … + (x — 1)k = (x + 1)l + … + (y — 1)l for fixed positive integers k and l. In this paper, we prove some effective and ineffective finiteness statements for the balancing numbers, using certain Baker-type Diophantine results and Bilu—Tichy theorem, respectively.

Hyperbolic Pascal triangles

In this paper, we introduce a new generalization of Pascals triangle. The new object is called the hyperbolic Pascal triangle since the mathematical background goes back to regular mosaics on the hyperbolic plane. We precisely describe the procedure of how to obtain a given type of hyperbolic Pascal triangle from a mosaic. Then we study certain quantitative properties such as the number, the sum, and the alternating sum of the elements of a row. Moreover, the pattern of the rows, and the appearance of some binary recurrences in a fixed hyperbolic triangle are investigated.

Lucas Diophantine Triples

Abstract In this paper, we show that the only triple of positive integers a < b < c such that ab + 1, ac + 1 and bc + 1 are all members of the Lucas sequence (Ln ) n≥0 is (a, b, c) = (1, 2, 3).

Diophantine equations with binary recurrences associated to Brocard–Ramanujan problem

In this paper, applying the Primitive Divisor Theorem, we solve completely the diophantine equation Gn1Gn2 · · ·Gnk + 1 = G 2 m in the positive integers k, m and n1 < n2 < · · · < nk if the binary recurrence {Gn}n=0 is either the Fibonacci sequence, or the Lucas sequence, or the sequence of balancing numbers. In case of Fibonacci numbers our result is a generalization of the theorem of Marques on the equation FnFn+1 · · ·Fn+k−1 + 1 = F 2 m. AMS Subject Classification Numbers: .

Only finitely many Tribonacci Diophantine triples exist

Abstract Diophantine triples taking values in recurrence sequences have recently been studied quite a lot. In particular the question was raised whether or not there are finitely many Diophantine triples in the Tribonacci sequence. We answer this question here in the affirmative. We prove that there are only finitely many triples of integers 1 ≤ u < v < w such that uv + 1, uw + 1, vw + 1 are Tribonacci numbers. The proof depends on the Subspace theorem.

On the Diophantine Equations (2 n − 1)(6 n − 1) = x2 and (a n − 1)(a kn − 1) = x2

In this paper we prove that the equation (2n − 1)(6n − 1) = x2 has no solutions in positive integers n and x. Furthermore, the equation (an − 1) (akn − 1) = x2 in positive integers a > 1, n, k > 1 (kn > 2) and x is also considered. We show that this equation has the only solutions (a,n,k,x) = (2,3,2,21), (3,1,5,22) and (7,1,4,120).

Power sums in hyperbolic Pascal triangles

Abstract In this paper, we describe a method to determine the power sum of the elements of the rows in the hyperbolic Pascal triangles corresponding to –4, q} with q ≥ 5. The method is based on the theory of linear recurrences, and the results are demonstrated by evaluating the kth power sum in the range 2 ≤ k ≤ 11.

Balancing Diophantine triples with distance 1

For a positive real number

Balancing with binomial coefficients

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