Euler's first proposition says that a number which is a STS, iff is divided by another prime STS, will give a quotient that is a STS. My question is, why is there a need for that primality? If I divide by a composite STS, will I still get a STS? I'm not the best in number theory but I sort of see it like this bear with me
$xy=STS$ where $x= prime STS$
STS can be made up of = non-STS * non-STS OR STS can be made up of = STS * STS
NON-STS made up of = non-STS * STS
Knowing that,
if I multiple a non-STS by a STS I will always have a non-STS left over, which in turns mean that that itself is not a STS. $xy (STS) = x (STS) * y (non-STS)$ is just not possible regardless of whether $x$ is a prime STS or not. If someone could help me out please. Thank you.
A natural number $m$ is a STS iff in its prime factorization each prime of the form $4k+3$ appears to an even power (possibly the $0$ power if that prime does not appear in the factorization). Now suppose $n=ab$ where $n$ is a STS, and assume also that $a$ is a STS. Then in the factorization of $n$ the $4k+3$ primeseach appear to even powers, and the assumption on $a$ gives that the factorization of it also contains each $4k+3$ prime to an even power. This forces even powers on the $4k+3$ primes in the factorization of $b,$ and so $b$ is also a STS.
I think maybe Euler's proposition was just trying to show the prime STS are primes in the ring of Gaussian integers.