My observation is that every odd semiprime can be written as the difference of two even perfect squares, divided by 4, or, in other words, in order to locate--and possibly factorize--odd semiprimes, we have the congruences $e_{i}^2 – e_{k}^2 ≡ 0~(mod~4)$, $ e_{i}^2 – e_{k}^2 ≡ 0~(mod~p)$ and $e_{i}^2 – e_{k}^2 ≡ 0~(mod~q)$, with $e_{i}~>~e_{k}$ being even integers, and $e_{k}$ possibly also being $0$.
My line of reasoning is the following.
Theorem 1:
In the "trivial" case, already known to and geometrically proven by the Pythagoreans, all odd numbers $o$ can be written as the difference of two squares of integers n, which is $o = (n+1)^2 – n^2$ with $o = 2n + 1$, i.e. the odd number $o$ equals the sum of the bases of the two squares.
Theorem 2:
Every odd semiprime can be written as the difference of two perfect squares - apart from the trivial case for all odd numbers. - Note that the following has been known to Pierre de Fermat (Fermat's factorization method).
Proof:
Let $p q$ be an odd semiprime, with its (non-trivial) odd prime factors $p$ and $q ≥ p$.
Then, due to the 1st and 2nd Binomial Theorems, we have
$(p + q)^2 = p^2 + 2 p q + q^2$
and
$(q~–~p)^2 = q^2 – 2 p q + p^2$.
Then, solving both equations for $2 p q$ , we have
$(p + q)^2~-~p^2~-~q^2 = 2 p q$
and
$q^2~+~p^2 - (q~–~p)^2 = 2 p q$.
In order to obtain $p q$ , we divide both equations by 4 and add them:
,
while by cancellation of terms we obtain
,
which is equivalent to
$(p~+~q)^2 + (q~–~p)^2 = 4 pq$.
Since both the sum and the difference of two odd primes must be even, this should finish the proof that every odd semiprime can be written as the difference of two even perfect squares, divided by 4■.
Regarding this, I have three questions: (1) Are there other ways to prove this, for example by means of quadratic residues or the Chinese Remainder Theorem? (2) Is this connected with Goldbach's Conjecture (i.e. if below any even perfect square > 4 there is at least one odd semiprime for which the congruence holds)? (3) If so, is it a reasonable pathway to a proof of Goldbach's conjecture to try and obtain a relation between the distribution of the odd semiprimes in relation to the distribution of the even perfect squares?