It has been shown that the behaviour of primordial gas collapsing in a dark matter minihalo can depend on the adopted choice of 3-body H$_2$ formation rate. The uncertainties in this rate span two orders of magnitude in the current literature, and so it remains a source of uncertainty in our knowledge of population III star formation. Here we investigate how the amount of fragmentation in primordial gas depends on the adopted 3-body rate. We present the results of calculations that follow the chemical and thermal evolution of primordial gas as it collapses in two dark matter minihalos. Our results on the effect of 3-body rate on the evolution until the first protostar forms agree well with previous studies. However, our modified version of GADGET-2 SPH also includes sink particles, which allows us to follow the initial evolution of the accretion disc that builds up on the centre of each halo, and capture the fragmentation in gas as well as its dependence on the adopted 3-body H$_2$ formation rate. We find that the fragmentation behaviour of the gas is only marginally effected by the choice of 3-body rate co-efficient, and that halo-to-halo differences are of equal importance in affecting the final mass distribution of stars.
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