On a class of combinatorial diophantine equations
نویسندگان
چکیده
We give a combinatorial proof for a second order recurrence for the polynomials pn(x), where pn(k) counts the number of integer-coordinate lattice points x = (x1, . . . , xn) with ‖x‖ = ∑n i=1|xi| ≤ k. This is the main step to get finiteness results on the number of solutions of the diophantine equation pn(x) = pm(y) if n and m have different parity. The combinatorial approach also allows to extend the original diophantine result to more general combinatorial situations.
منابع مشابه
A Generalized Fibonacci Sequence and the Diophantine Equations $x^2pm kxy-y^2pm x=0$
In this paper some properties of a generalization of Fibonacci sequence are investigated. Then we solve the Diophantine equations $x^2pmkxy-y^2pm x=0$, where $k$ is positive integer, and describe the structure of solutions.
متن کاملCombinatorial Diophantine Equations and a Refinement of a Theorem on Separated Variables Equations
We look at Diophantine equations arising from equating classical counting functions such as perfect powers, binomial coefficients and Stirling numbers of the first and second kind. The proofs of the finiteness statements that we give use a variety of methods from modern number theory, such as effective and ineffective tools from Diophantine approximation. As a tool for one part of the statement...
متن کاملDiophantine Equations Related with Linear Binary Recurrences
In this paper we find all solutions of four kinds of the Diophantine equations begin{equation*} ~x^{2}pm V_{t}xy-y^{2}pm x=0text{ and}~x^{2}pm V_{t}xy-y^{2}pm y=0, end{equation*}% for an odd number $t$, and, begin{equation*} ~x^{2}pm V_{t}xy+y^{2}-x=0text{ and}text{ }x^{2}pm V_{t}xy+y^{2}-y=0, end{equation*}% for an even number $t$, where $V_{n}$ is a generalized Lucas number. This pape...
متن کاملOn the Decidability of Diophantine Problems in Combinatorial Geometry
In spite of Matiyasevic's solution to Hubert's 10th problem some fifteen years ago it is still unknown whether there exists an algorithm to decide the solvability of diophantine equations within the field of rational numbers. In this note we show the equivalence of this problem with a conjecture of B. Grünbaum [6] on rational coordinatizability in combinatorial geometry. Such an algorithm exist...
متن کاملClass Numbers of Quadratic Fields Determined by Solvability of Diophantine Equations
In the literature there has been considerable attention given to the exploration of relationships between certain diophantine equations and class numbers of quadratic fields. In this paper we provide criteria for the insolvability of certain diophantine equations. This result is then used to determine when related real quadratic fields have class number bigger than 1. Moreover, based on criteri...
متن کامل