  (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.
