A 38 Integers 15 ( 2015 ) Polynomial Sequences on Quadratic Curves
نویسندگان
چکیده
In this paper we generalize the study of Matiyasevich on integer points over conics, introducing the more general concept of radical points. With this generalization we are able to solve in positive integers some Diophantine equations, relating these solutions by means of particular linear recurrence sequences. We point out interesting relationships between these sequences and known sequences in OEIS. We finally show connections between these sequences and Chebyshev and Morgan-Voyce polynomials, finding new identities.
منابع مشابه
EEH: AGGH-like public key cryptosystem over the eisenstein integers using polynomial representations
GGH class of public-key cryptosystems relies on computational problems based on the closest vector problem (CVP) in lattices for their security. The subject of lattice based cryptography is very active and there have recently been new ideas that revolutionized the field. We present EEH, a GGH-Like public key cryptosystem based on the Eisenstein integers Z [ζ3] where ζ3 is a primitive...
متن کاملElliptic Curves and Continued Fractions
We detail the continued fraction expansion of the square root of the general monic quartic polynomial. We note that each line of the expansion corresponds to addition of the divisor at infinity, and interpret the data yielded by the general expansion. The paper includes a detailed ’reminder exposition’ on continued fractions of quadratic irrationals in function fields. A delightful ‘essay’ [16]...
متن کاملThe Large Sieve Inequality for Quadratic Polynomial Amplitudes
Basic examples of sparse sequences of integers are provided by the sequence of values of polynomials of degree ≥ 2 with integer coefficients. The present article is concerned with the case when polynomial is of degree 2. Indeed, in a recent note, Liangyi Zhao [4], showed, by an elegant application of the double large sieve inequality of Bombieri and Iwaniec, that one has the estimate given belo...
متن کاملRepresentation of Integers by Near Quadratic Sequences
Following a statement of the well-known Erdős-Turán conjecture, Erdős mentioned the following even stronger conjecture: if the n-th term an of a sequence A of positive integers is bounded by αn2, for some positive real constant α, then the number of representations of n as a sum of two terms from A is an unbounded function of n. Here we show that if an differs from αn 2 (or from a quadratic pol...
متن کاملQuadratic Form Representations via Generalized Continuants
H. J. S. Smith proved Fermat’s two-square theorem using the notion of palindromic continuants. In this paper we extend Smith’s approach to proper binary quadratic form representations in some commutative Euclidean rings, including rings of integers and rings of polynomials over fields of odd characteristic. Also, we present new deterministic algorithms for finding the corresponding proper repre...
متن کامل