We analyze the two brightest Chandra X-ray flares detected from Sagittarius A*, with peak luminosities more than 600 x and 245 x greater than the quiescent X-ray emission. The brightest flare has a distinctive double-peaked morphology --- it lasts 5.7 ksec ($sim 2$ hours), with a rapid rise time of 1500 sec and a decay time of 2500 sec. The second flare lasts 3.4 ksec, with rise and decay times of 1700 sec and 1400 sec. These luminous flares are significantly harder than quiescence: the first has a power law spectral index $Gamma = 2.06pm 0.14$ and the second has $Gamma = 2.03pm 0.27$, compared to $Gamma = 3.0pm0.2$ for the quiescent accretion flow. These spectral indices (as well as the flare hardness ratios) are consistent with previously-detected Sgr A* flares, suggesting that bright and faint flares arise from similar physical processes. Leveraging the brightest flares long duration and high signal-to-noise, we search for intraflare variability and detect excess X-ray power at a frequency of $ u approx 3$ mHz, but show that it is an instrumental artifact and not of astrophysical origin. We find no other evidence (at the 95% confidence level) for periodic or quasi-periodic variability in either flares time series. We also search for non-periodic excess power but do not find compelling evidence in the power spectrum. Bright flares like these remain our most promising avenue for identifying Sgr A*s short timescale variability in the X-ray, which may probe the characteristic size scale for the X-ray emission region.
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