We investigate the galaxy overdensity around proto-cluster scale quasar pairs at high (z>3) and low (z~1) redshift based on the unprecedentedly wide and deep optical survey of the Hyper Suprime-Cam Subaru Strategic Program (HSC-SSP). Using the first-year survey data covering effectively ~121 deg^2 with the 5sigma depth of i~26.4 and the SDSS DR12Q catalog, we find two luminous pairs at z~3.3 and 3.6 which reside in >5sigma overdense regions of g-dropout galaxies at i<25. The projected separations of the two pairs are R_perp=1.75 and 1.04 proper Mpc, and their velocity offsets are Delta V=692 and 1448 km s^{-1}, respectively. This result is in clear contrast to the average z~4 quasar environments as discussed in Uchiyama et al. (2017) and implies that the quasar activities of the pair members are triggered via major mergers in proto-clusters, unlike the vast majority of isolated quasars in general fields that may turn on via non-merger events such as bar and disk instabilities. At z~1, we find 37 pairs with R_perp<2 pMpc and Delta V<2300 km s^{-1} in the current HSC-Wide coverage, including four from Hennawi et al. (2006). The distribution of the peak overdensity significance within two arcminutes around the pairs has a long tail toward high density (>4sigma) regions. Thanks to the large sample size, we find a statistical evidence that this excess is unique to the pair environments when compared to single quasar and randomly selected galaxy environments at the same redshift range. Moreover, there are nine small-scale (R_perp<1 pMpc) pairs, two of which are found to reside in cluster fields. Our results demonstrate that <2 pMpc-scale quasar pairs at both redshift range tend to occur in massive haloes, although perhaps not the most massive ones, and that they are useful to search for rare density peaks.
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