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

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 ∈ E ( K ) be a point of infinite order. It is natural to ask how many integers n ≥ 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 ∈ E ( ℚ ) as above, we show that there is at most one such n ≥ 3 .

On the Euler function of repdigits

Florian Luca (2008)

Czechoslovak Mathematical Journal

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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 b n - 1 b - 1 = y b m - 1 b - 1 , where x , y ∈ { 1 , ... , b - 1 } , has only finitely many positive integer solutions ( x , y , m , n ) .

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 ≥ 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 ≡ 0 ( mod q ) and use these bounds to show that lim sup n → ∞ P ( n ! + 2 n - 1 ) / n ≥ ( 2 π 2 + 3 ) / 18 .