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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

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 .

On the least common multiple of Lucas subsequences

Shigeki Akiyama, Florian Luca (2013)

Acta Arithmetica

We compare the growth of the least common multiple of the numbers u a 1 , . . . , u a n and | u a 1 u a n | , where ( u n ) n 0 is a Lucas sequence and ( a n ) n 0 is some sequence of positive integers.

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