Here, we show that there is no positive integer $n$ such that the $n$th Cullen number $C_n=n2^n+1$ has the property that it is composite but $\phi(C_n)\mid C_n-1$.

/pdf/cullen-numbers-with-the-lehmer-property-17qucoj7hw.pdf

Cullen Numbers with the Lehmer Property

Generalized Cullen numbers in linear recurrence sequences

The m-th Cullen number Cm is a number of the form m2 m + 1 and the m-th Woodall number Wm has the form m2 m 1. In 2003, Luca and Stuanicua proved that the largest Fibonacci number in the Cullen sequence is F4 = 3 and that F1 = F2 = 1 are the largest Fibonacci numbers in the Woodall sequence. Very recently, the second author proved that, for any given s > 1, the equation Fn = ms m ± 1 has only finitely many solutions, and they are effectively computable. In this note, we shall provide the explicit form of the possible solutions.

/pdf/fibonacci-s-cullen-and-s-woodall-numbers-2h7626e1m0.pdf

Fibonacci s-Cullen and s-Woodall Numbers

A Cullen number is a number of the form $m2^m+1$, where $m$ is a positive integer. In 2004, Luca and St\u anic\u a proved, among other things, that the largest Fibonacci number in the Cullen sequence is $F_4=3$. Actually, they searched for generalized Cullen numbers among some binary recurrence sequences. In this paper, we will work on higher order recurrence sequences. For a given linear recurrence $(G_n)_n$, under weak assumptions, and a given polynomial $T(x)\in \mathbb{Z}[x]$, we shall prove that if $G_n=mx^m+T(x)$, then \[ m\ll\log \log |x|\log^2(\log \log |x|)\ \mbox{and}\ n\ll\log |x|\log\log |x|\log^2(\log \log |x|), \] where the implied constant depends only on $(G_n)_n$ and $T(x)$.

Generalized Cullen Numbers in Linear Recurrence Sequences

For k ≥ 2, the k-generalized Fibonacci sequence (F
 n
 (k))
 n
 is defined by the initial values 0, 0,..., 0, 1 (k terms) and such that each term afterwards is the sum of the k preceding terms. In this paper, we find all generalized Fibonacci numbers written in the form 2
 a
 + 3
 b
 + 5
 c
 . This work generalizes a recent Marques-Togbe result [20] concerning the case k = 2.

Generalized Fibonacci numbers of the form 2 a + 3 b + 5 c

Using Selberg's sieve upper bounds for the number of primes of the formn2
 n
 +1 withn≤x and the number of primes of the formp2
 p
 +1 withp≤x are derived.

ber Primzahlen der Formn2 n +1 bzw.p2 p +1@@@On prime numbers of the formn2 n +1 respectivelyp2 p +1

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ber Primzahlen der Formn2 n +1 bzw.p2 p +1@@@On prime numbers of the formn2 n +1 respectivelyp2 p +1