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Both names are not used in modern Poland at all. Bezprim is known just from history (Polish Kingdom and Czech Kingdom) and Przyszek is known as surname for some modern Poles. It could have been used as first name in middle ages though.
Etymologicaly Przyszek is diminutive (-ek suffix) from Przysz, which in turn is diminutive (-sz suffix) from complex oldPolish names like Przybysław / Przybygniew / Przybyrad etc. from "przyby-" "to arrive, to come".
Bezprim is completely "black" and we don't even know the proper version (Bezprym or Bezprzem or Przezprzem etc).
Regards