BLAST (Balloon-borne Large-Aperture Submillimeter Telescope) performed the first deep and wide extragalactic survey at 250, 350 and 500 um. The extragalactic number counts at these wavelengths are important constraints for modeling the infrared galaxies evolution. [...] We use three methods to identify the submillimeter sources. 1) Blind extraction. [...] The photometry is computed with a new simple and quick PSF fitting routine (FASTPHOT). [...] 2) Extraction with prior. [...] 3) A stacking analysis. [...] With the blind extraction, we reach 97, 83 and 76 mJy at resp. 250, 350 and 500 um with a 95% completeness. With the prior extraction, we reach 76 mJy (resp. 63 and 49 mJy) at 250 um (resp. 350 and 500 um). With the stacking analysis, we reach 6.2 mJy (resp. 5.2 and 3.5 mJy) at 250 um (resp. 350 and 500 um). The differential submillimeter number counts are derived, and start showing a turnover at flux densities decreasing with increasing wavelength. There is a very good agreement with the P(D) analysis of Patanchon et al. (2009). At bright fluxes (>100 mJy), the Lagache et al. (2004) and Le Borgne et al. (2009) models slightly overestimate the observed counts, but there is a very good agreement near the peak of differential number counts. [...] Counts are available at: http://www.ias.u-psud.fr/irgalaxies/downloads.php
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