On the lcm of the differences of eight primes

Abstract:

Following C. A. Spiro, who has found eight primes for which \[ \operatorname {lcm}{({p_j} - {p_i})_{1 \leq i < j \leq 8}} = 5040,\] we show that for every set of eight odd primes $\{ {q_1}, \ldots ,{q_8}\}$, one has $5040|\operatorname {lcm}({q_j} - {q_i})$. Moreover, $\operatorname {lcm}({q_j} - {q_i}) = 5040$ infinitely often, under the assumption of the 8-tuple conjecture.