(abridged) We present results of a search for Lyman break galaxies (LBGs) at z ~ 5 in a 618 square-arcmin field including the HDF-N taken by Subaru Prime Focus Camera. Utilizing the published redshift data of the HDF-N and its flanking fields, the color selection criteria are chosen so that LBGs are picked out most efficiently and least contaminated by foreground objects. The numbers of LBG candidates detected are 310 in 23.0 < I_c < 25.5. The rest-frame UV luminosity function(LF) of LBGs at z ~ 5 is derived statistically. The fraction of contamination is estimated to be ~50% in the faintest magnitude range. The completeness of the survey is ~80% at the bright part of the sample, and ~20% in the faintest magnitude range (25.0 < I_c <= 25.5). The LF of LBG candidates at z ~ 5 does not show a significant difference from those at z ~ 3 and 4, though there might be a slight decrease in the fainter part. The UV luminosity density within the observational limit is 0.56 - 0.69 times smaller than that obtained for LBGs at z ~ 3, depending on the adopted cosmology and the integration range of the LF. The similarity of the LFs at redshifts 5 to 3 implies that most of LBGs at z ~ 5 should have faded out at z ~ 3 and LBGs at z \~ 5 are different galaxies from those seen at z ~ 3, if we take face values for ages of the LBGs at z ~ 3 obtained by the SED fitting in which a continuous star formation in an individual galaxy is assumed. However, if the star formation in LBGs is sporadic, the similarity of the LF at z ~ 3 and 5 would be explained. Such sporadic star formation has been suggested by hydrodynamical simulations and semi-analytic models with collisional starbursts, and the trend of the cosmic star formation history predicted by these studies resembles to that estimated from the UV luminosity density within the observational limit.