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Let P 1 , P 2 , ... , P l ∈ Z X , Y be homogeneous polynomials of degree 2, with nonpositive discriminant and let Q be a polynomial with integer coefficients. Under the assumption that 0 is a root of Q with odd multiplicity, we show that the variant of the Brocard–Ramanujan Diophantine equation Q n ! = ∏ i = 1 l P i X i , Y i α i has only finitely many integer solutions n , X ¯ l , Y ¯ l , α ¯ l , where X ¯ l = X 1 , X 2 , ... X l . [ABSTRACT FROM AUTHOR]

In this note we give exact formulas (and asymptotics) for the number of rational points of bounded height on weighted projective stacks over global function fields. [ABSTRACT FROM AUTHOR]

A Diophantine m-tuple over a finite field F q is a set { a 1 , ... , a m } of m distinct elements in F q ∗ such that a i a j + 1 is a square in F q whenever i ≠ j . In this paper, we study M(q), the maximum size of a Diophantine tuple over F q , assuming the characteristic of F q is fixed and q → ∞ . By explicit constructions, we improve the lower bound on M(q). In particular, this improves a recent result of Dujella and Kazalicki by a multiplicative factor. [ABSTRACT FROM AUTHOR]

We explore a generalization of the Markov numbers that is motivated by a specific generalized cluster algebra arising from an orbifold, in the sense of Chekhov and Shapiro. We give an explicit algorithm for computing these generalized Markov numbers and exhibit several patterns analogous to those that appear within the ordinary Markov numbers. Along the way, we present formulas related to continued fractions and snake graphs. [ABSTRACT FROM AUTHOR]

Let S be a finite, fixed set of primes. In this paper, we show that the set of integers c which have at least two representations as a difference between a factorial and an S-unit is finite and effectively computable. In particular, we find all integers that can be written in at least two ways as a difference of a factorial and an S-unit associated with the set of primes { 2 , 3 , 5 , 7 } . [ABSTRACT FROM AUTHOR]

Let d ≥ 2 be an integer which is not a square. We show that if (L n) n ≥ 0 is the Lucas sequence and (X m , Y m) m ≥ 1 is the mth solution of the Pell equation X 2 - d Y 2 = ± 1 , then the equation Y m = L n has at most two positive integer solutions (m, n) except for d = 2 when it has the three solutions (m , n) = (1 , 1) , (2 , 0) , (5 , 7) . [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]

A Huff curve over a field K is an elliptic curve defined by the equation ax(y2-1)=by(x2-1) where a,b∈K are such that a2≠b2. In a similar fashion, a general Huff curve over K is described by the equation x(ay2-1)=y(bx2-1) where a,b∈K are such that ab(a-b)≠0. In this note we express the number of rational points on these curves over a finite field Fq of odd characteristic in terms of Gaussian hypergeometric series 2F1(λ):=2F1ϕϕϵ|λ where ϕ and ϵ are the quadratic and trivial characters over Fq, respectively. Consequently, we exhibit the number of rational points on the elliptic curves y2=x(x+a)(x+b) over Fq in terms of 2F1(λ). This generalizes earlier known formulas for Legendre, Clausen and Edwards curves. Furthermore, using these expressions we display several transformations of 2F1. Finally, we present the exact value of 2F1(λ) for different λ's over a prime field Fp extending previous results of Greene and Ono. [ABSTRACT FROM AUTHOR]

We establish upper bounds for the number of primitive integer solutions to inequalities of the shape $$0<|F(x, y)| \le h$$ , where $$F(x , y) =(\alpha x + \beta y)^r -(\gamma x + \delta y)^r \in \mathbb {Z}[x ,y]$$ , $$\alpha $$ , $$\beta $$ , $$\gamma $$ and $$\delta $$ are algebraic constants with $$\alpha \delta -\beta \gamma \ne 0$$ , and $$r \ge 5$$ and h are integers. As an important application, we pay special attention to binomial Thue's inequalities $$|ax^r - by^r| \le c$$ . The proofs are based on the hypergeometric method of Thue and Siegel and its refinement by Evertse. [ABSTRACT FROM AUTHOR]

Under certain conditions on the coefficients, we derive asymptotic formulas for the number of representations of zero and of large positive integers by the cubic forms that can be written as $L_{1}(x_{1},x_{2},x_{3}) Q_{1}(x_{1},x_{2},x_{3})+ L_{2}(x_{4},x_{5},x_{6}) Q_{2}(x_{4},x_{5}, x_{6}) + a_{7} x_{7}^{3}$, where L and L are linear forms, Q and Q are quadratic forms, and a is a nonzero integer. [ABSTRACT FROM AUTHOR]