Thank you to Zach for sending us some very interesting findings on primes, inspired by Vicky Neale's talk 7 Things You Need to Know About Prime Numbers.
He begins by looking at a related problem on Hilbert numbers (that is, numbers that are 1 more than a multiple of 4) and looks at what primes would be if only Hilbert numbers existed. Note that an s-prime is a Hilbert number that cannot be divided by a smaller Hilbert number (other than 1).
He then looks at how prime factorisation behaves differently with s-primes.
Afterwards, he uses these ideas to look for patterns in primes by developing a sieve similar to the ones used in Vicky's talk.
Here are Zach's detailed findings.