We present results from our X-ray analysis of the first systematic search for triple AGN in nearby (z<0.077) triple galaxy mergers. We analyze archival Chandra observations of 7 triple galaxy mergers with BAYMAX (Bayesian Analysis of Multiple AGN in X-rays), fitting each observation with single, dual, and triple X-ray point source models. In doing so, we conclude that 1 triple merger has one X-ray point source (SDSS J0858+1822, although its unlikely to be an AGN); 5 triple mergers are likely composed of two X-ray point sources (NGC 3341, SDSS J1027+1749, SDSS J1631+2352, SDSS J1708+2153, and SDSS J2356$-$1016); and one system is composed of three X-ray point sources (SDSS J0849+1114). By fitting the individual X-ray spectra of each point source, we analyze the 2-7 keV luminosities as well as the levels of obscuration associated with each potential AGN. We find that 4/5 dual X-ray point source systems have primary and secondary point sources with bright X-ray luminosities (L_2-7 kev >10^40 erg s^-1), possibly associated with 4 new undetected dual AGN. The dual and triple point source systems are found to have physical separations between 3-9 kpc and flux ratios between 2x10^-3 - 0.84. A multi-wavelength analysis to determine the origin of the X-ray point sources discovered in this work is presented in our companion paper (Foord et al. 2020c).
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