TITLE: The truth of Nyonyon Theorem
QUESTION [5 upvotes]: Let $s= ab$ be a semiprime number. Then the Nyonyon Theorem states that $s+a$, $s+b$, $s+a+b$ are not all coprime to three.
(In other words: there exist no $s= ab$ semiprimes such that $s+a$, $s+b$ and $s+a+b$ are coprime to three.)
I've checked plenty of numbers, and I think Nyonyon Theorem is absolutely true, but I don't understand why this phenomena happens. Can someone explain/proof to me why this thing happens ?

REPLY [2 votes]: Assume the existence of such a semiprime $s=ab$,we have $s+b=b(a+1)$ and $s+a=a(b+1)$ are all coprime to $3$ hence $a\equiv b\equiv 1 \mod 3$ but in this case $s+a+b\equiv 1+1+1 \equiv 0\mod 3 $ contradiction