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We find all solutions of three exponential Diophantine equations, arising from certain quadratic, cubic and quartic identities. The first identity comes from a painting of the famous Russian painter Nikolay Bogdanov-Belsky, highlighted by Ja. I. Perelman. The equations have five, four and six terms, respectively, so they cannot be handled by classical tools based upon Baker's method. To solve the equations we use our method developed earlier, which is based upon Skolem's conjecture, local considerations and a computational approach.
Bertók et al. (Mon,) studied this question.