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I know by definition, a semi-prime has factors that are prime numbers. But what I'm unsure of, is if there is ever a case where the product of two prime numbers results in number with factors OTHER than the original two prime numbers?

Or will any product of two primes ALWAYS be only divisible by 1, itself, and the original two primes (or I guess the original prime if you squared it).

Eddie
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It is indeed always a semiprime. Think about it this way:

Consider a number $x = qq_1$ for some primes $q$ and $q_1$. We know that if $p$ is a prime and $p|ab$, then $p|a$ or $p|b$. Well, what if some other prime $p|x$? This would imply that $p|q$ or $p|q_1$, which is not possible because $q$ and $q_1$ are themselves prime. Therefore, the only possible nontrivial divisors of $x$ are $q$ and $q_1$.

Kaj Hansen
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  • Could you explain why " We know that if p is a prime and p|ab, then p|a or p|b" , please ? Also if this works for 2 primes q and q1 such that x =qq1, then it should work for products of more than 2 primes shouldnt it ? – Manu de Hanoi Jul 20 '20 at 10:39
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    " We know that if p is a prime and p|ab, then p|a or p|b" , is proved here : https://en.wikipedia.org/wiki/Euclid%27s_lemma – Manu de Hanoi Jul 20 '20 at 11:04