منابع مشابه
Arithmetic Progressions and Pellian Equations
We consider arithmetic progressions on Pellian equations x2 − d y2 = m, i.e. integral solutions such that the y-coordinates are in arithmetic progression. We construct explicit infinite families of d,m for which there exists a five-term arithmetic progression in the y-coordinate, and we prove the existence of infinitely many pairs d,m parametrized by the points of an elliptic curve of positive ...
متن کاملDiophantine approximation and Diophantine equations
The first course is devoted to the basic setup of Diophantine approximation: we start with rational approximation to a single real number. Firstly, positive results tell us that a real number x has “good” rational approximation p/q, where “good” is when one compares |x − p/q| and q. We discuss Dirichlet’s result in 1842 (see [6] Course N◦2 §2.1) and the Markoff–Lagrange spectrum ([6] Course N◦1...
متن کاملSolvability of Diophantine Equations
Attila Bérczes (University of Debrecen): On arithmetic properties of solutions of norm form equations. Abstract. Let α be an algebraic number of degree n and K := Q(α). Consider the norm form equation NK/Q(x0 + x1α+ x2α + . . .+ xn−1α) = b in x0, . . . , xn−1 ∈ Z. (1) Let H denote the solution set of (1). Arranging the elements of H in an |H| × n array H, one may ask at least two natural questi...
متن کاملComplete decomposition of Dickson-type polynomials and related Diophantine equations
We characterize decomposition over C of polynomials f (a,B) n (x) defined by the generalized Dickson-type recursive relation (n ≥ 1), f (a,B) 0 (x) = B, f (a,B) 1 (x) = x, f (a,B) n+1 (x) = xf (a,B) n (x)− af (a,B) n−1 (x), where B, a ∈ Q or R. As a direct application of the uniform decomposition result, we fully settle the finiteness problem for the Diophantine equation f (a,B) n (x) = f (â,B̂)...
متن کاملFamilies of Diophantine equations
This is a report on the recent work by Claude Levesque and the author on families of Diophantine equations. This joint work started in 2010 in Rio, and this is still work in progress. The lecture in Lahore on March 11, 2013 was mainly devoted to a survey of results on Diophantine equations, with the last part dealing with some recent results. Here we describe the content of the recent joint pap...
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ژورنال
عنوان ژورنال: Journal of Number Theory
سال: 2011
ISSN: 0022-314X
DOI: 10.1016/j.jnt.2011.02.005