Let S be a fixed set of primes and let a 1 , . . . , a m be positive distinct integers. We call the m-tuple (a 1 , . . . , a m ) S-Diophantine, if for all i = j the integers aiaj + 1 = si,j are S-integers. In this paper we show that if |S| = 2, then under some technical restrictions no S-Diophantine quadruple exists.
In this paper, we introduce a new generalization of Pascal's triangle. The new object is called the hyperbolic Pascal triangle since the mathematical background goes back to regular mosaics on the hyperbolic plane. We describe precisely 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 appearence of some binary recurrences in a fixed hyperbolic triangle are investigated.
The positive integer x is a (k, l)-balancing number for y (x y − 2) if 1 k + 2 k + • • • + (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.
Let S denote a set of primes and let a 1 , . . . , am be positive distinct integers. We call the m-tuple (a 1 , . . . , am) an S-Diophantine tuple if a i a j +1 = s i,j are S-units for all i = j. In this paper, we show that no S-Diophantine quadruple (i.e. m = 4) exists if S = {2, q} with q ≡ 3 (mod 4) or q < 10 9 . For two arbitrary primes p, q < 10 5 we gain the same result.2010 Mathematics Subject Classification. 11D61,11D45.
Abstract. In this paper, we show that there are no three distinct positive integers a, b, c such that ab + 1, ac + 1, bc + 1 are all three Fibonacci numbers.
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