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Hayat Bensella

Publications and source records attributed to Hayat Bensella.

5 recordsLinked to original sources
arXiv2026

A Korselt Characterization of Carmichael Numbers of the Form $(aC_n+1)(aC_{n+1}+1)(aC_{n+2}+1)$

We study products \[ N=(aC_n+1)(aC_{n+1}+1)(aC_{n+2}+1), \] where $(C_n)$ is the Lucas--balancing sequence. Under the assumption that all three factors are prime, we give a necessary and sufficient condition for $N$ to be a Carmichael number: this happens precisely when $n$ is even and $a=kC_{n+1}$ for a positive integer $k$ satisfying \[ 48k\equiv 7\pmod{C_nC_{n+2}}. \] We derive further congruence restrictions, an explicit parametrization of the integer solutions of the resulting congruence, a lower bound for $k$, and a divisibility relation involving the middle prime factor. The characterization is conditional on the primality of the three factors and does not assert that such prime triples exist for infinitely many parameters.

math.NT↗
arXiv2026

On Carmichael Numbers Constructed from Consecutive Jacobsthal Numbers

We prove that no Carmichael number has the form \[ N=(2^mJ_n+1)(2^mJ_{n+1}+1)(2^mJ_{n+2}+1), \] where \[ J_n=\frac{2^n-(-1)^n}{3} \] is the Jacobsthal sequence, \(m,n\ge1\), and all three factors are prime. Assuming that such an \(N\) is Carmichael, Korselt's criterion yields divisibility conditions relating consecutive Jacobsthal numbers to powers of two. Combining these conditions with the Binet formula and a finite case analysis for \(n\le4\), we obtain a contradiction. Consequently, this family contains no Carmichael numbers.

math.NT↗
arXiv2026

A Note on Carmichael Numbers of the form $ N=(d+1)(3d+1)(4d+1)$

We study integers of the form \[ N=(d+1)(3d+1)(4d+1). \] Using Korselt's criterion, we prove that, provided the three factors are prime, $N$ is a Carmichael number if and only if \[ d\equiv4\pmod{12}. \] We also show that the special case $d=2^m$ produces only $1105$ under the prime-factor hypothesis.

math.GM↗
arXiv2023

$k$-Fibonacci numbers which are Narayana's cows numbers

In this paper,we find all generalized Fibonacci numbers which are Narayana's cows numbers. In our proofs, we use both Baker's theory of nonzero linear forms in logarithms of algebraic numbers and the Baker-Davenport reduction method.

math.NT↗