A Note on squares in arithmetic progressions, II

Enrico Bombieri; Umberto Zannier

Displaying similar documents to “A Note on squares in arithmetic progressions, II”

Small exponent point groups on elliptic curves

Florian Luca, James McKee, Igor E. Shparlinski (2006)

Journal de Théorie des Nombres de Bordeaux

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Let $E$ be an elliptic curve defined over $F_{q}$ , the finite field of $q$ elements. We show that for some constant $η > 0$ depending only on $q$ , there are infinitely many positive integers $n$ such that the exponent of $E (F_{q^{n}})$ , the group of $F_{q^{n}}$ -rational points on $E$ , is at most $q^{n} exp (- n^{η / log log n})$ . This is an analogue of a result of R. Schoof on the exponent of the group $E (F_{p})$ of $F_{p}$ -rational points, when a fixed elliptic curve $E$ is defined over $ℚ$ and the prime $p$ tends to infinity.

Note on special arithmetic and geometric means

Horst Alzer (1994)

Commentationes Mathematicae Universitatis Carolinae

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We prove: If $A (n)$ and $G (n)$ denote the arithmetic and geometric means of the first $n$ positive integers, then the sequence $n \mapsto n A (n) / G (n) - (n - 1) A (n - 1) / G (n - 1)$ $(n \geq 2)$ is strictly increasing and converges to $e / 2$ , as $n$ tends to $\infty$ .

On arithmetic progressions on Edwards curves

Enrique González-Jiménez (2015)

Acta Arithmetica

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Let $m \in ℤ_{> 0}$ and a,q ∈ ℚ. Denote by $_{m} (a, q)$ the set of rational numbers d such that a, a + q, ..., a + (m-1)q form an arithmetic progression in the Edwards curve $E_{d} : x ² + y ² = 1 + d x ² y ²$ . We study the set $_{m} (a, q)$ and we parametrize it by the rational points of an algebraic curve.

Product sets cannot contain long arithmetic progressions

Dmitrii Zhelezov (2014)

Acta Arithmetica

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Let B be a set of complex numbers of size n. We prove that the length of the longest arithmetic progression contained in the product set B.B = bb’ | b,b’ ∈ B cannot be greater than O((nlog²n)/(loglogn)) and present an example of a product set containing an arithmetic progression of length Ω(nlogn). For sets of complex numbers we obtain the upper bound $O (n^{3 / 2})$ .

Herbrand consistency and bounded arithmetic

Zofia Adamowicz (2002)

Fundamenta Mathematicae

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We prove that the Gödel incompleteness theorem holds for a weak arithmetic Tₘ = IΔ₀ + Ωₘ, for m ≥ 2, in the form Tₘ ⊬ HCons(Tₘ), where HCons(Tₘ) is an arithmetic formula expressing the consistency of Tₘ with respect to the Herbrand notion of provability. Moreover, we prove $T ₘ ⊬ H C o n s^{I ₘ} (T ₘ)$ , where $H C o n s^{I ₘ}$ is HCons relativised to the definable cut Iₘ of (m-2)-times iterated logarithms. The proof is model-theoretic. We also prove a certain non-conservation result for Tₘ.

Linear independence of linear forms in polylogarithms

Raffaele Marcovecchio (2006)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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For $x \in ℂ$ , $| x | < 1$ , $s \in ℕ$ , let ${Li}_{s} (x)$ be the $s$ -th polylogarithm of $x$ . We prove that for any non-zero algebraic number $α$ such that $| α | < 1$ , the $ℚ (α)$ -vector space spanned by $1, {Li}_{1} (α), {Li}_{2} (α), \dots$ has infinite dimension. This result extends a previous one by Rivoal for rational $α$ . The main tool is a method introduced by Fischler and Rivoal, which shows the coefficients of the polylogarithms in the relevant series to be the unique solution of a suitable Padé approximation problem.