# proving the fundamental theorem of arithmetic

The book is trying to prove the fundamental theorem of arithmetic but does he mean by saying that at least one of the inequality must be true with strict inequality ? And from where did he come up with this last inequality ?

• They cannot both be equalities, since otherwise $p^2=n=p'^2$ which is impossible with $p\neq p'$. Now we have, say $p\leq\sqrt{n}, p'<\sqrt{n}$, so $pp'<n$. – Wojowu Apr 17 at 22:03

Since $$p\ne p'$$, you cannot have both $$n=p^2$$ and $$n=p'^2$$. Thus, say, $$n>p'^2$$.
If we multiply the inequalities, we get $$n^2>p^2p'^2$$, so $$n>pp'$$.