As a tiny matter of terminology, "prime number theorem" is usually reserved for the theorem on the asymptotic number of primes (https://en.wikipedia.org/wiki/Pr

As a tiny matter of terminology, "prime number theorem" is usually reserved for the theorem on the asymptotic number of primes (https://en.wikipedia.org/wiki/Prime_number_theorem https://en.wikipedia.org/wiki/Prime_number_theorem). I would probably just say "by prime factorisation", or, if you want to be really precise, "by Euclid's lemma" (https://en.wikipedia.org/wiki/Euclid%27s_lemma https://en.wikipedia.org/wiki/Euclid%27s_lemma).