Just to add something to jsf's list, you can only downgrade a PSP with firmware between 2.81 and 3.03 inclusive is if you have a copy of GTA: LCS with the 2.00 update on the disc. Firmwares 1.00 to 2.80 inclusive can run homebrew without a copy of that disc.

(Homebrew is basically everything not released on UMD.)

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