Search

We investigate the number of squares in a very broad family of binary recurrence sequences with u 0 = 1. We show that there are at most two distinct squares in such sequences (the best possible result), except under very special conditions where we prove there are at most three such squares. [ABSTRACT FROM AUTHOR]

Given a minimal model of an elliptic curve, E / K , over a finite extension, K , of Q p for any rational prime, p , and any point P ∈ E (K) of infinite order, we determine precisely min ⁡ (v (ϕ n (P)) , v (ψ n 2 (P))) , where v is a normalised valuation on K and ϕ n (P) and ψ n (P) are polynomials arising from multiplication by n on this model of the curve. [ABSTRACT FROM AUTHOR]

In 2016, Jang and Kim stated a conjecture about the norms of indecomposable integers in real quadratic number fields Q (D) where D > 1 is a squarefree integer. Their conjecture was later disproved by Kala for D ≡ 2 mod 4. We investigate such indecomposable integers in greater detail. In particular, we find the minimal D in each congruence class D ≡ 1 , 2 , 3 mod 4 that provides a counterexample to the Jang-Kim Conjecture; provide infinite families of such counterexamples; and state a refined version of the Jang-Kim Conjecture. Lastly, we prove a slightly weaker version of our refined conjecture that is of the correct order of magnitude, showing the Jang-Kim Conjecture is only wrong by at most O (D). [ABSTRACT FROM AUTHOR]

For all integers m ≥ n ≥ 2 , we exhibit infinite families of purely periodic Jacobi–Perron Algorithm (JPA) expansions of dimension n with period length equal to m along with the associated Hasse–Bernstein units. Some observations on the units of Levesque–Rhin as well as the periodicity of the JPA expansion of ( m 1 / 3 , m 2 / 3 ) are also made. [ABSTRACT FROM AUTHOR]

In this paper, we sharpen and simplify our earlier results based on Thue's Fundamentaltheorem and use it to obtain effective irrationality measures for certain roots of particular polynomials of the form ( x − t ) n + ( x + t ) n , where n ≥ 4 is a positive integer and t is a negative integer. For n = 4 and n = 5 , we find infinitely many such algebraic numbers. [ABSTRACT FROM AUTHOR]

Abstract: For every number field , with , we show that the number of non-associates of the same norm in a full module in does not depend only on , but can also depend on the module itself. As a corollary, the same can be true for the number of families of solutions of degenerate norm form equations. So the uniform bound obtained by Schmidt for the number of solutions in the non-degenerate case does not hold always here. For three-variable norm forms not arising from full modules, we do obtain a Schmidt-type bound for the number of families of solutions that, together with the above result, completes this aspect of the study of three-variable norm forms. [Copyright &y& Elsevier]