I noticed that in my Rome game, it's 525 and my technologies are all at 1, while Egypt and Bosporian kingdom have 6 or 7. I have taken omens that increase the research points and I have appointed researchers with good traits. What am I doing wrong?
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Citizens are the strata of population which brings you research. As Rome you conquer a lot of foreign culture provinces and enslave the conquered citizens. As a result the percentage of citizens (optimally 30%) in your population drops and causes your relative tech progress lagging.
To remedy this encourage the freemen to citizens conversion by adopting the "Citizenship" idea (+7% conversion rate) and possibly enact the "Lex Gellia et Cornelia" (+2% conv. rate, +10% tax). As freemen form the pool for the citizen promotion you can encourage them too with the idea of "Emancipation".
Colonies also help your tech progress somehow, although they have low civilization rating the citizen population quick grow there more than compensates that.
But I tag switched to Bosporan Kingdom, and they only gain +2.1 from citizens. It seems that for them to reach the next tech level, they only need 315 investment, while I need over 10 times more (around 3600).
Exactly, the higher total population you have the higher the Technology Cost grows. So it's not the absolute value of RPs your citizens produce but the proportion of citizens among your population what actually counts.
The Exact Technology Research Points Cost Formula
With around 30% citizens you get a technology advance really close to or exactly in it's intended year of introduction, one tech advance per 10 years. With more positive modifiers, like great civil magistrate and prime minister, academies, papyrus trade, philosophies you can compensate for lower citizen proportion, or you can even get the technology of about 8-10 years in advance. The additional RP penalty for getting technology in advance grows pretty steep soon though.
The Exact Technology Research Points Cost Formula
Thanks, but... the Romans were known for their high-tech military. I'm now at level 3 while Egypt and other nations are at 7 or 8. It's hard to beat them, and I'll never unlock some buildings.
Another question, why do my factions' loyalty lower when I execute a character that is taken prisoner from another nation?
You could always try to start a civil war in Rome, depending on the circumstances (how much territory and armies the rebels obtain) you could shrink your country, and your tech requirements, and catch up to the competition while you send the rebels to their deaths.
If you annex another country and capture occupants of their nation, those characters will gain faction conviction just like your normal Rome characters do. So they'll be placed in a faction appropriate for them if their country had been a Republic (or stay in the same one if you just annexed a Republic). So executing those people WILL adversely affect your nation's factions, even if the character isn't native to your faction. It's a bit of a weird gameplay issue, one I'm not sure if Paradox sees as a bug or not, but it's mostly there to prevent you from willy-nilly executing people and escaping any longterm consequences unscathed. The only person you're allowed to legally execute (and happens automatically if they're caught) is a civil war Rebel Leader.
If you annex another country and capture occupants of their nation, those characters will gain faction conviction just like your normal Rome characters do. So they'll be placed in a faction appropriate for them if their country had been a Republic (or stay in the same one if you just annexed a Republic). So executing those people WILL adversely affect your nation's factions, even if the character isn't native to your faction. It's a bit of a weird gameplay issue, one I'm not sure if Paradox sees as a bug or not, but it's mostly there to prevent you from willy-nilly executing people and escaping any longterm consequences unscathed. The only person you're allowed to legally execute (and happens automatically if they're caught) is a civil war Rebel Leader.