Abstract
Let ⊘ denote Euler’ totient function Consider the three equations k⊘(n) = n, k⊘(n) = n+1, and k⊘(n) = n-1. The first equation has been completely solved. The two latter equations are referred to as Lehmer' equations, and neither of these has been solved. We initiate the study of three ring-theoretic problems which generalize the preceding number-theoretic equations. Ultimately, we solve these three problems for the class of local rings with identity.
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