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a-ae abuts aliquot sequences arithmetic assume atoms axioms bytes choose component COMPUTING SCIENCE conditions are equivalent contains continuum Corollary cyclic D.H. Lehmer define definition denote Department of Mathematics Diophantine direct sum direct summand disjoint driver element elementarily equivalent endomorphism ring equations exists factors Fibonacci numbers finitely generated submodule free groups free product Guy and J.L. Hence homeomorphism idempotent inertia theorems integral manifold intersects invariant set J.L. Selfridge Jordan-Kette left ideal Lemma length function lifted modulo main sequences Math MATHEMATICS STATISTICS matrix neighborhood orbits perfect number polynomial positive integer predicate prime projective cover projective module projective summand Proof properties Proposition prove radAf radM regular modules regular ring relay Research Paper result ring with identity semi semiperfect ring semiregular module semiregular ring STATISTICS AND COMPUTING subset Suppose surface of section Theorem 1.5 theory of free transition chain UNIVERSITY OF CALGARY Ware 11 x-ex