On Carmichael Numbers Constructed from Consecutive Jacobsthal Numbers

Source abstract

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.