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 ? enter image description here

enter image description here

  • 2
    $\begingroup$ 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 $pp'<n$. $\endgroup$ – 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 $n>pp'$.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.