Primitive divisors of Lucas and Lehmer sequences, II

Paul M. Voutier

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Multiplicative dependence in number fields

J. Loxton, Alfred van der Poorten (1983)

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A quantitative primitive divisor result for points on elliptic curves

Patrick Ingram (2009)

Journal de Théorie des Nombres de Bordeaux

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Let $E / K$ be an elliptic curve defined over a number field, and let $P \in E (K)$ be a point of infinite order. It is natural to ask how many integers $n \geq 1$ fail to occur as the order of $P$ modulo a prime of $K$ . For $K = ℚ$ , $E$ a quadratic twist of $y^{2} = x^{3} - x$ , and $P \in E (ℚ)$ as above, we show that there is at most one such $n \geq 3$ .

Remarks on number theory I. On primitive α-abundant numbers

P Erdös (1959)

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The divisors of integers II

R. Hall (1975)

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On the Euler function of repdigits

Florian Luca (2008)

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For a positive integer $n$ we write $φ (n)$ for the Euler function of $n$ . In this note, we show that if $b > 1$ is a fixed positive integer, then the equation $φ (x \frac{b^{n} - 1}{b - 1}) = y \frac{b^{m} - 1}{b - 1}, where x, y \in {1, ..., b - 1},$ has only finitely many positive integer solutions $(x, y, m, n)$ .

Euler indicators of binary recurrence sequences.

Florian Luca (2002)

Collectanea Mathematica

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On the largest prime factor of $n! + 2^{n} - 1$

Florian Luca, Igor E. Shparlinski (2005)

Journal de Théorie des Nombres de Bordeaux

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For an integer $n \geq 2$ we denote by $P (n)$ the largest prime factor of $n$ . We obtain several upper bounds on the number of solutions of congruences of the form $n! + 2^{n} - 1 \equiv 0 (mod q)$ and use these bounds to show that $\underset{n \to \infty}{lim sup} P (n! + 2^{n} - 1) / n \geq (2 π^{2} + 3) / 18 .$

Displaying similar documents to “Primitive divisors of Lucas and Lehmer sequences, II”

Multiplicative dependence in number fields

A quantitative primitive divisor result for points on elliptic curves

Remarks on number theory I. On primitive α-abundant numbers

The divisors of integers II

On the Euler function of repdigits

Euler indicators of binary recurrence sequences.

On the largest prime factor of n ! + 2 n - 1

On the largest prime factor of $n! + 2^{n} - 1$