The proof of a conjecture of Graham for sequences containing primes

Abstract:

Let ${a_1} < {a_2} < \cdots < {a_n}$ be a finite sequence of positive integers. R. L. Graham has conjectured that ${\max _{i,j}}\left \{ {{a_i}/({a_i},{a_j})} \right \} \geqslant n$. We verify this conjecture in case at least one of the ${\alpha _i}$’s is prime.