Let $A$ be a set of positive integers satisfying the following properties: (i) if $m$ and $n$ belong to $A$, then $m+n$ belong to $A$; (ii) there is no prime number that divides all elements of $A$.
(a) Suppose $n_1$ and $n_2$ are two integers belonging to $A$ such that $n_2-n_1 >1$. Show that you can find two integers $m_1$ and $m_2$ in $A$ such that $0< m_2-m_1 < n_2-n_1$
(b) Hence show that there are two consecutive integers belonging to $A$.
(c) Let $n_0$ and $n_0+1$ be two consecutive integers belonging to $A$. Show that if $n\geq n_0^2$ then $n$ belongs to $A$.