(One post up, wow) With no offense to frojoe or Sushi Warrior, a buck, buck-boost or constant current source is not a realistic project for someone who needs help calculating the value of a resistor for a string of LEDs. EDIT: Saw your edit. Again, incase you didn't read the original question (why post?) the design requirement was three strings of 20 LEDs at 20mA each. I don't know where 5 volts will get you, but supposing you rigged up a 3.5v boost circuit with 100% efficiency one AA battery will not supply the 2.8 amps at 1.5 volts required to run such a string.