I think you're on the right track... but for the benefit of others reading this thread, in hopes of mastering the fundamentals involved...
a) Current is ALWAYS important... about on the level that "money" is "important" in business! (^_^)
b) Theory is all well and good... in theory. And a good starting point. But you often have to temper it with the results of tests. So, in your scenario, yes, I'd start with the 80s. But.
If you make the resistor too "big", the LED will be too dim, or even off, as you surmised. Make the resistor too small, and enough current will flow that you damage the LED or the Arduino, or both. If the specs spoke of the LED operating on 20mA, that probably means that 20mA is a sensible level to use. You could probably push it a bit for an even brighter light, at the cost of increasing the change of a premature (up to "instant"!) burnout.
It's not just the LEDs limits which have to be considered. Remember to pay attention to the limits on what current can be run through an individual Arduino pin, AND the limits on the safe total current, when the pins' currents are combined, too. You may need to drive transistors with the Arduino, and use THEM to drive the LEDs.
Onward!
Hook up your LED, through a decent ammeter. ("Decent" adequate. Doesn't have to be "five star")
Look and see what ACTUAL current you get with your resistor. As I said... theory is fine in theory.
Once you've made that start, you are in a position to "tweak" the resistor value you're using with that particular batch of LEDs, if necessary.
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Further analysis....
In the original post, we had....
(5-3.4)/20ma = 80
Here's what TauTau was doing....
With LEDs, Ohm's Law doesn't "work" in a simple way.
The "5" was the voltage applied at the "top" of the circuit.
The 3.4 taken from it was the "voltage drop" assumed for the LED. That number is typical, but for some LEDs you need to use a slightly different number, to have things come out "right".
The 20 was the current he wanted to see flowing, based on what he'd been told about what would be "right" for that particular (model of) LED.
After doing the arithmetic, out came 80. That said "you need an 80 ohm resistor".
Within the limits of the currents which a given LED was made to operate on, the "voltage drop" number for a given LED (and the others with the same part number, in the case of LEDs from a sensible supplier) doesn't change. Not with different "overall" applied voltage. Not when the current flowing changes, which it will, if the "overall" voltage is changed.
I hope the above helps someone see what the early parts of this thread were saying.