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In this paper we analytically study the recurrence equations of an Isingmodel with three competing interactions on a Cayley tree of order three. Weexactly describe paramagnetic and ferromagnetic phases of the Ising model. Weobtain some rigorous results: critical temperatures and curves, number ofphases, partition function. Ganikhodjaev et al. [J. Concrete and ApplicableMathematics, 9 (1), 26-34 (2011)] have numerically studied the Ising model on asecond-order Cayley tree. We compare the numerical results to exact solutionsof mentioned model.

Exact description of paramagnetic and ferromagnetic phases of an Ising model on a third-order Cayley tree