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0. Introduction and preliminaries. The Liouville and Bôcher theorems are not a new subject. We found over 150 papers about the Liouville theorem for solutions of various (mostly elliptic) differential equations. The Bôcher theorem is somewhat less popular. Besides the classical case of harmonic functions, we found [4] and [5], where the Bôcher theorem was proved for solutions of elliptic equations ([4]) and some quasielliptic system of equations ([5]). There is also [3], where the Bôcher theorem for solutions of the Laplace–Beltrami equation was studied. While reading the excellent book of Axler, Bourdon and Ramey [2], we discovered that one can prove the Liouville and Bôcher theorems for polyharmonic functions, using the facts and ideas on harmonic functions given in that book. We must add only some elementary calculations. Our proofs will be on the calculus level. We start with the following 0.1. Definition. Let Ω be an open set in R. A function u : Ω → R is m-polyharmonic iff ∆u = 0 on Ω, where ∆u = ∆(∆(. . . ∆u)) (m times), and ∆u = ∑n j=1 ∂ u/∂xj is the Laplace operator. Let us next recall

Elementary proofs of the Liouville and Bôcher theorems for polyharmonic functions