In this first paper in the SUPER GOODS series on powerfully star-forming galaxies in the two GOODS fields, we present a deep SCUBA-2 survey of the GOODS-N at both 850 and 450 micron (central rms noise of 0.28 mJy and 2.6 mJy, respectively). In the central region the 850 micron observations cover the GOODS-N to near the confusion limit of ~1.65 mJy, while over a wider 450 arcmin^2 region---well complemented by Herschel far-infrared imaging---they have a median 4-sigma limit of 3.5 mJy. We present >4-sigma catalogs of 186 850 micron and 31 450 micron selected sources. We use interferometric observations from the SMA and the VLA to obtain precise positions for 114 SCUBA-2 sources (28 from the SMA, all of which are also VLA sources). We present new spectroscopic redshifts and include all existing spectroscopic or photometric redshifts. We also compare redshifts estimated using the 20 cm to 850 micron and the 250 micron to 850 micron flux ratios. We show that the redshift distribution increases with increasing flux, and we parameterize the dependence. We compute the star formation history and the star formation rate (SFR) density distribution functions in various redshift intervals, finding that they reach a peak at z=2-3 before dropping to higher redshifts. We show that the number density per unit volume of SFR>500 solar mass per year galaxies measured from the SCUBA-2 sample does not change much relative to that of lower SFR galaxies from UV selected samples over z=2-5, suggesting that, apart from changes in the normalization, the shape in the number density as a function of SFR is invariant over this redshift interval.
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