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X-ray source based on the Thomson scattering of laser interacting with energetic electron beams features high photon energy, small spot size, and good collimation. However, the photon number is insufficient for practical application because of the small cross section of the Thomson scattering. To solve this problem, here, we replace a traditional Gaussian laser pulse with a flying focus laser pulse to extend interaction length and restrain nonlinear effects. Simulation results show that the scattered photon number can be increased by about 25 and 2 times for high and low energy lasers, respectively. In particular, a 1010 photon number can be generated with a 10 J flying focus laser pulse, and the energy spread can also be greatly reduced for high energy laser, from a broad spectrum to a monoenergetic peak. Combining these two advantages, the peak spectrum brightness of x ray is 3 × 108 photons/keV at 240 keV, which is about three orders of magnitude more than the traditional case. [ABSTRACT FROM AUTHOR]