We use the correspondence between Kloosterman sums and elliptic curves to prove identities for binary and ternary Kloosterman sums. The identities follow from rational parametrizations of modular curves X0(l). In the binary case we give the results from an unpublished manuscript by Mika R.S. Kojo and in the ternary case we give new results. We show that two identities published by other authors are consequences of identities proved in this paper.
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