Polychromatic solid-state lamps that produce white light by additive mixing of the emissions from primary colored light emitting diodes (LEDs) should have a higher luminous efficiency that those using phosphors. These lamps require emission spectra that feature an optimal trade-off between luminous efficacy and color rendering. We developed a mathematical technique that allows us to maximize the luminous efficacy and general color rendering index (CRI) for the white solid-state lamp composed of an arbitrary number of primary LEDs with given spectra. We use this method in order to compare the optimal efficacy and general CRI for 4 and 5 primary LEDs with that for 2 and 3 LEDs. For a particular color temperature, the required number of primary LEDs depends on the trade-off between efficacy and general CRI. The quadrichromatic lamp is shown to meet requirements for most practical applications. Quintichromatic lamps and lamps with a higher number of primary LEDs yield negligible benefit in improving color rendering. However, quintichromatic LED lamps are capable of producing quasi-continuous spectra that might meet special lighting needs.© (2001) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.