[tbm57]

I might be misunderstanding what they accomplished here - doesn't this mean that oscillators needed to be found for all prime number periods?

for me, random coincidence that this popped up today, I just published a little holiday-themed GoL variant last week here: https://52games52weeks.com/gameofchristmas

[kuhewa]

No, because increasing the width or height of this loop by one cell increases the period by one.

https://conwaylife.com/?rle=36b2o$35bobo$29b2o4bo$27bo2bo2b2...

[abetusk]

From the abstract:

""" ... At the turn of the millennium, only twelve oscillator periods remained ... . The search has finally ended, with ... the final two periods, 19 and 41, ... """

Note that 19 and 41 are prime.

I'm not familiar with this line of research in the GoL (nor most others) but I assume that it was proved all prime periods (above 41, say) have been known since the turn of the century, or thereabouts.

I suspect what happened is that there were "easy" constructions for large periods that could be done. I would think that once you have the freedom of large periods, you can construct large gadgets and thus prove with "relative ease" that you can get any large prime number period.

My bet is that smaller periods are harder because of the smallness restriction.

[cultureswitch]

I think that beyond a certain period you have so much logic at your disposal that you can construct two organisms sending messages to each other and vary the distance in increments of one to get any period you want.

[lotharbot]

You're basically correct, though it's usually done with 4 mechanisms making an adjustable glider loop, which can have any period of 43 or more just by increasing the spacing: https://conwaylife.com/wiki/P43_Snark_loop

[Paul-Craft]

Right. If you want to have an oscillator with period $pq$ where $p$ and $q$ are primes, then you can just use independent oscillators with period $p$ to one with period $q$. They'd already discovered a construction for $p \geq 61,$ and most of the values below that, so only 19 and 41 were left.

[DerekL]

You also need all powers of primes.