In this paper, we study the Diophantine equation bk+(a+ b)k+(2a+ b)k++(a(x-1) +b)k=y(y +c)(y +2c)(y +(& ell;-1) c) , where a, b, c, k, & ell; are given integers under natural conditions. We prove some effective results for special values for c, k and & ell;, and obtain a general ineffective result based on the Bilu-Tichy method.

We prove effective finiteness results concerning polynomial values of the sums b^k +( a+b) ^k + ⋯ + ( a( x-1) + b) ^k and b^k - ( a+b) ^k + ( 2a+b) ^k - … + (-1)^x-1( a( x-1) + b) ^k, where a 0,b, k are given integers with (a,b)=1 and k ≥ 2 .

Motivated by the work of David Singmaster, we study the number of times an integer can appear among the Stirling numbers of both kinds. We provide an upper bound for the occurrences of all the positive integers, and present certain questions for further study. Some numerical results and conjectures concerning the related diohantine equations are collected.

Diophantine problems related to binomial coefficients have a vast literature. In this paper we investigate power and polynomial values of row sums of hyperbolic Pascal triangles, which are recently introduced generalizations of Pascal's classical triangle. We prove various effective and ineffective finiteness results.

We investigate Diophantine problems concerning linear combinations of polynomials of the shape $$a_0 x+a_1x(x+1)+ a_2x(x+1)(x+2)+ \cdots + a_n x(x+1)\ldots (x+n)$$ with $$n\in \mathbb{N}\cup\{0\}$$.We provide effective finiteness results for the power, shifted power, and quadratic polynomial values of these linear combinations, generalizing the analogous results of Hajdu, Laishram and Tengely [10], and of Bérczes, Hajdu, Luca and the author [2] given for the sums $$x+x(x+1)+x(x+1)(x+2)+ \cdots + x(x+1)\ldots (x+n)$$, i.e., for the case $$a_0=a_1= \cdots = a_n = 1$$. Our work is closely connected also with some results of Tengely and Ulas [15] concerning the case when the coefficients $$a_0,a_1, \ldots, a_n$$ are zeroes and ones.

We investigate the power and polynomial values of the polynomials P-n (X) = Pi(n)(k=)(0) (X2 - 3k - X-3k - 1) for n is an element of N. We prove various ineffective and effective finiteness results. in the case 0 <= n <= 3, we determine all pairs x, y of integers such that P-n(x) = y(2) or P-n(x) = y(3).

We investigate polynomial values of sums of products of consecutive integers. For the degree two case we give effective finiteness results, while for the higher degree case we provide ineffective finiteness theorems. For the latter purpose, we also show that the polynomials corresponding to the sums of products we investigate, are indecomposable.

We prove ineffective finiteness results on the integer solutions x , y of the equations b^k + (a + b)^k + ⋯ + (a(x - 1) + b)^k = g(y) and b^k - (a + b)^k + (2a + b)^k - ⋯ + (-1)^x-1(a(x - 1) + b)^k = g(y), where g(x) ∈ℚ[x], deg g(x) ≧ 3, and a ≠ 0, b are given integers with gcd( a , b ) = 1.

We consider the Diophantine equation \(P_n (x) = g(y)\) in \(x, y\) where \(P_n (x), g(x) \in \mathbb {Q}[x], \deg g(x) \ge 3\) and \(\left\{ P_n (x)\right\} _{n \ge 0}\) is an Appell sequence. Under some reasonable assumptions on \(P_n(x)\) we prove an ineffective finiteness result on the above equation.

We show that the alternating power sum r(n) - (m + r)(n) + (2m + r)(n) - . . . + (-1)(l-1) ((l-1) m + r)(n) can be expressed in terms of Stirling numbers of the first kind and r-Whitney numbers of the second kind. We also prove a necessary and sufficient condition for the integrality of the coefficients of the polynomial extensions of the above alternating power sum.

In this paper, we consider the Diophantine equationb(k) + (a + b)(k) + ... + (a (x - 1) + b)(k) == d(l) + (c + d)(l) + ... + (c (y - 1) + d)(l),where a, b, c, d, k, l are given integers with gcd (a, b) = gcd (c, d) = 1, k not equal l. We prove that, under some reasonable assumptions, the above equation has only finitely many solutions.

In an attempt to present a refinement of Faulhaber’s theorem concerning sums of powers of natural numbers, the authors investigate and derive all the possible decompositions of the polynomial Sa,bk(x) which is given by Sa,bk(x)=bk+(a+b)k+(2a+b)k+⋯+(a(x−1)+b)k.

In Part I (cf. [13]) of this paper, the title equation was solved in x, y, n ∈ Z with |xy| > 1, n ≥ 3 for a collection of positive integers A, B, C under certain bounds.In the present paper we extend these results to much larger ranges of A, B, C. We give among other things all the solutions for A = C = 1, B < 235 (cf.Theorem 1), and for C = 1, A, B ≤ 50, with six explicitly given exceptions (A, B, n) (cf.Theorem 3).The equations under consideration are solved by combining powerful techniques, including Frey curves and associated modular forms, lower bounds for linear forms in logarithms, the hypergeometric method of Thue and Siegel, local methods, classical cyclotomy and computational approaches to Thue equations of low degree.Along the way, we derive a new result on the solvability of binomial Thue equations (cf.Theorem 6) which is crucial in the proof of our Theorems 1 and 2. Some important applications of our theorems will be given in a forthcoming paper.

Year: 2010 Vol.: 77 Fasc.: 3-4 Title: On binomial Thue equations and ternary equations with S-unit coe±cients Author(s): Andr¶as Bazs¶o In this paper we obtain some new results for a collection of equations of the form (2) AxniByn = §1 resp. (3) AxniByn = zm with m 2 f3; ng, where x, y, z, A, B, n are unknown nonzero integers such that n ¸ 3 is a prime and AB is composed of two ¯xed primes. We prove among other things that under certain conditions formulated in Section 2, equations (3) have no solutions with jxyj > 1, Ax, By and z coprime and n > 13 (cf. Theorems 2 to 4). Combining this with some other results and techniques, we establish a similar result for equations (2) (cf. Theorem 1). Address: Andr¶as Bazs¶o Institute of Mathematics Number Theory Research Group of the Hungarian Academy of Sciences University of Debrecen H-4010 Debrecen, P.O. Box 12 Hungary

In Part I of this paper, the title equation was solved in x;y;n 2Z with jxyj > 1; n ‚ 3 for a collection of positive integers A, B, C under certain bounds. In the present paper we extend these results to much larger ranges of A, B, C. We give among other things all the solutions for A = C = 1, B < 235 (cf. Theorem 1), and for C = 1, A;B • 50, with six explicitly given exceptions (A;B;n) (cf. Theorem 3). The equations under consideration are solved by combining powerful techniques, including Frey curves and associated modular forms, lower bounds for linear forms in logarithms, the hypergeometric method of Thue and Siegel, local methods, classical cyclotomy and computational approaches to Thue equations of low degree. Along the way, we derive a new result on the solvability of binomial Thue equations (cf. Theorem 6) which is crucial in the proof of our Theorems 1 and 2. Some important applications of our theorems will be given in a forthcoming paper.

In Part I (cf. [13]) of this paper, the title equation was solved in x, y, n is an element of Z with vertical bar xy vertical bar > 1, n >= 3 for a collection of positive integers A, B, C under certain bounds. In the present paper we extend these results to much larger ranges of A, B, C. We give among other things all the solutions for A = C = 1, B < 235 (cf. Theorem 1), and for C = 1, A, B <= 50, with six explicitly given exceptions (A, B, n) (cf. Theorem 3). The equations tinder consideration are solved by combining powerful techniques, including Frey curves and associated modular forms, lower bounds for linear forms in logarithms, the hypergeometric method of Thue and Siegel, local methods, classical cyclotomy and computational approaches to Thue equations of low degree. Along the way, we derive a new result oil the solvability of binomial Thue equations (cf. Theorem 6) which is crucial in the proof of our Theorems I and 2. Some important applications of our theorems will be given in a forthcoming paper.

In this paper we present our computational experiences on those special solutions of the norm form equation NK/Q(x0 + x1α + · · · + xn−1αn−1) = 1 for which x0, . . . , xn−1 ∈ Z are consecutive elements of an arithmetic progression.