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Let { P n } n ≥ 0 be the sequence of Pell numbers defined by P 0 = 0 , P 1 = 1 and P n + 2 = 2 P n + 1 + P n for all n ≥ 0 and let { Q n } n ≥ 0 be its companion sequence, the Pell–Lucas numbers defined by Q 0 = Q 1 = 2 and Q n + 2 = 2 Q n + 1 + Q n for all n ≥ 0 . In this paper, we find all Pell and Pell–Lucas numbers which can be written as difference of two repdigits. It is shown that the largest Pell and Pell–Lucas numbers which can be written as difference of two repdigits are P 6 = 70 = 77 - 7 and Q 7 = 478 = 555 - 77. [ABSTRACT FROM AUTHOR]

The integer sequence defined by P n + 3 = P n + 1 + P n with initial conditions P 0 = 1 and P 1 = P 2 = 0 is known as the Padovan sequence P n n ∈ Z . A recurrence sequence { u n } n ∈ Z is said to be of Padovan-type if it satisfies the same recurrence relation as the Padovan sequence but with arbitrary initial values u 0 , u 1 , u 2 not all zero. The most famous Padovan-type sequence is the Perrin sequence given by u 0 = 3 , u 1 = 0 and u 2 = 2 . We show that every Padovan-type sequence has at most 2 zeros, except for nonzero multiples of shifts of the Padovan sequence which has 0-multiplicity 5. We also show that P n n ∈ Z has total multiplicity 62, i.e., there are 62 pairs (m , n) ∈ Z 2 with m < n for which P m = P n . As a consequence, we found that { P n } n ∈ Z has multiplicity 9, being 1 the most repeated Padovan number. Finally, we prove that the Perrin sequence has exactly 1 zero, total multiplicity 23 and multiplicity 4. [ABSTRACT FROM AUTHOR]

Let (Pn)n≥0 be the sequence of Padovan numbers defined by P0 = 0, P1 = 1 = P2, and Pn+3 = Pn+1 +Pn for all n ≥ 0. In this paper, we find all Padovan numbers that are concatenations of two distinct repdigits. [ABSTRACT FROM AUTHOR]

We give an upper bound on the number of extensions of a triple to a quadruple for the Diophantine m-tuples with the property D(4). We also confirm the conjecture of the uniqueness of such an extension in some special cases. [ABSTRACT FROM AUTHOR]

A set of m distinct positive integers { a 1 , ... a m } is called a D (q) - m -tuple for nonzero integer q if the product of any two increased by q, a i a j + q , i ≠ j is a perfect square. Due to certain properties of the sequence, there are many D(q)-Diophantine triples related to the Fibonacci numbers. A result of Baćić and Filipin characterizes the solutions of Pellian equations that correspond to D(4)-Diophantine triples of a certain form. We generalize this result in order to characterize the solutions of Pellian equations that correspond to D(l2)-Diophantine triples satisfying particular divisibility conditions. Subsequently, we employ this result and bounds on linear forms in logarithms of algebraic numbers in order to classify all D(9) and D(64)-Diophantine triples of the form { F 2 n + 8 , 9 F 2 n + 4 , F k } and { F 2 n + 12 , 16 F 2 n + 6 , F k } , where F i denotes the ith Fibonacci number. [ABSTRACT FROM AUTHOR]

Let { X ℓ } ℓ ≥ 1 be the sequence of X-coordinates of the positive integer solutions (X, Y) of the Pell equation X 2 - d Y 2 = ± 1 corresponding to a nonsquare integer d > 1 . We show that there are only a finite number of nonsquare integers d > 1 such that there are at least two different elements of the sequence { X ℓ } ℓ ≥ 1 that can be represented as a linear combination of prime powers with fixed primes and coefficients, restricted to the condition that the exponent of the largest prime is the greatest of all exponents. Moreover, we solve explicitly the case in which two of the X-coordinates above are a sum of a power of two and a power of three under the above condition on the exponents. This work is motivated by the recent paper Bertók et al. (Int J Number Theory 13(02):261–271, 2017). [ABSTRACT FROM AUTHOR]

Let {Bn}n≥0 be the sequence of Balancing numbers defined by B0 = 0, B1 = 1, and Bn+2 = 6Bn+1 − Bn for all n ≥ 0. In this paper, we find all repdigits in base 10 which can be written as a sum of three Balancing numbers. [ABSTRACT FROM AUTHOR]

Let { F n } n ≥ 0 be the sequence of Fibonacci numbers defined by F 0 = 0 , F 1 = 1 and F n + 2 = F n + 1 + F n for all n ≥ 0 . In this paper, for an integer d ≥ 2 which is square-free, we show that there is at most one value of the positive integer x participating in the Pell equation x 2 - d y 2 = ± 4 which is a sum of two Fibonacci numbers, with a few exceptions that we completely characterize. [ABSTRACT FROM AUTHOR]

For an integer k≥2, let {Fn(k)}n≥0 be the k-generalized Fibonacci sequence which starts with 0,…,0,1 (k terms) and each term afterwards is the sum of the k preceding terms. In this paper, we find all integers c having at least two representations as a difference between a k-generalized Fibonacci number and a power of 2 for any fixed k≥4. This paper extends previous work from Ddamulira et al. (Proc Math Sci 127(3): 411-421, 2017. 10.1007/s12044-017-0338-3) for the case k=2 and Bravo et al. (Bull Korean Math Soc 54(3): 069-1080, 2017. 10.4134/BKMS.b160486) for the case k=3. [ABSTRACT FROM AUTHOR]

Dujella and Pethő, generalizing a result of Baker and Davenport, proved that the set { 1 , 3 } cannot be extended to a Diophantine quintuple. As a consequence of our main result, we show that the Diophantine pair { 1 , b } is regular if b − 1 is a prime power. [ABSTRACT FROM AUTHOR]

In this paper, we consider the $$D(\pm k)$$ -triple $$\{k\mp 1,k, 4k\mp 1\}$$ and we prove that, if k is not a perfect square then: This extends a work done by Fujita [13]. [ABSTRACT FROM AUTHOR]

Let ( F n ) n ≥ 0 be the Fibonacci sequence given by F m + 2 = F m + 1 + F m , for m ≥ 0 , where F 0 = 0 and F 1 = 1 . In 2011, Luca and Oyono proved that if F m s + F m + 1 s is a Fibonacci number, with m ≥ 2 , then s = 1 or 2. A well-known generalization of the Fibonacci sequence, is the k -generalized Fibonacci sequence ( F n ( k ) ) n which is defined by the initial values 0 , 0 , … , 0 , 1 ( k terms) and such that each term afterwards is the sum of the k preceding terms. In this paper, we generalize Luca and Oyono's method by proving that the Diophantine equation ( F m ( k ) ) s + ( F m + 1 ( k ) ) s = F n ( k ) has no solution in positive integers n , m , k and s , if 3 ≤ k ≤ min ⁡ { m , log ⁡ s } . [ABSTRACT FROM AUTHOR]